Displaying 1-10 of 29 results found.
3, 1, 3, -1390, -139895, 2064875400, 999912530925, -128585633463727440, -176876516433064573125, 109242473594498195269718400, 333170810414553853376721961875, -698025623281503752808511373154720000, -4073023833462008382211035330291042675375
COMMENTS
Coefficients in Stirling's asymptotic expansion of the Gamma function, normalized to integers using factor 3*(2*n)!^2/n!!.
FORMULA
a(n) = 3*(2*n)!*(6*n+1)!!/(n!!*4^n) * Sum_{i=0..2*n} Sum_{j=0..i} Sum_{k=0..j} (-1)^k*2^i*k^(2*n+i+j)*C(2*n,i)*C(i,j)*C(j,k) / ((2*n+2*i+1)*(2*n+i+j)!), assuming 0^0 = 1 (when n = 0), n!! = A006882(n), C(n,k) = A007318(n,k) are binomial coefficients.
MAPLE
h := proc(k) option remember; local j; `if`(k=0, 1,
(h(k-1)/k-add((h(k-j)*h(j))/(j+1), j=1..k-1))/(1+1/(k+1))) end:
g := n -> doublefactorial(2*n-1)*(2*n)!^2/doublefactorial(n):
MATHEMATICA
Table[3 (2n)!^2/n!! (6n+1)!!/4^n Sum[(-1)^m 2^k StirlingS2[2n+k+m, m]/((2n+2k+1) (2n+k+m)! (2n-k)! (k-m)!), {k, 0, 2n}, {m, 0, k}], {n, 0, 12}]
Stirling's formula: numerators of asymptotic series for Gamma function.
(Formerly M5400 N2347)
+10
29
1, 1, 1, -139, -571, 163879, 5246819, -534703531, -4483131259, 432261921612371, 6232523202521089, -25834629665134204969, -1579029138854919086429, 746590869962651602203151, 1511513601028097903631961, -8849272268392873147705987190261, -142801712490607530608130701097701
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 257, Eq. 6.1.37.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 267, #23.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
FORMULA
The coefficients c_k have g.f. 1 + Sum_{k >= 1} c_k/z^k = exp( Sum_{k >= 1} B_{2k}/(2k(2k-1)z^(2k-1)) ).
Numerators/denominators: A001163(n)/ A001164(n) = (6*n+1)!!/(4^n*(2*n)!) * Sum_{i=0..2*n} Sum_{j=0..i} Sum_{k=0..j} (-1)^k*2^i*k^(2*n+i+j)*C(2*n,i) *C(i,j)*C(j,k)/((2*n+2*i+1)*(2*n+i+j)!), assuming 0^0 = 1 (when n = 0), n!! = A006882(n), C(n,k) = A007318(n,k) are binomial coefficients. - Vladimir Reshetnikov, Nov 05 2015 Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence
c_0 = 1, c_n = (1/n) * Sum_{k=0..n-1} B_{n-k+1}*c_k/(n-k+1) for n > 0.
a(n) is the numerator of c_n. (End)
EXAMPLE
Gamma(z) ~ sqrt(2*Pi) * z^(z-1/2) * e^(-z) * (1 + 1/(12*z) + 1/(288*z^2) - 139/(51840*z^3) - 571/(2488320*z^4) + ... ), z -> infinity in |arg z| < Pi.
MAPLE
h := proc(k) option remember; local j; `if`(k=0, 1,
(h(k-1)/k-add((h(k-j)*h(j))/(j+1), j=1..k-1))/(1+1/(k+1))) end:
StirlingAsympt := proc(n) option remember; h(2*n)*2^n*pochhammer(1/2, n) end:
A001163 := n -> numer(StirlingAsympt(n));
MATHEMATICA
Numerator[ Reverse[ Drop[ CoefficientList[ Simplify[ PowerExpand[ Normal[ Series[n!, {n, Infinity, 17}]]Exp[n]/(Sqrt[2Pi n]*n^(n - 17))]], n], 1]]]
(* Second program: *)
h[k_] := h[k] = If[k==0, 1, (h[k-1]/k-Sum[h[k-j]*h[j]/(j+1), {j, 1, k-1}]) / (1+1/(k+1))];
StirlingAsympt[n_] := h[2n]*2^n*Pochhammer[1/2, n];
a[n_] := StirlingAsympt[n] // Numerator;
PROG
(PARI) a(n)=local(A, m); if(n<1, n==0, A=vector(m=2*n+1, k, 1); for(k=2, m, A[k]=(A[k-1]-sum(i=2, k-1, i*A[i]*A[k+1-i]))/(k+1)); numerator(A[m]*m!/2^n/n!)) /* Michael Somos, Jun 09 2004 */ (Sage)
@cached_function
def h(k):
if k<=0: return 1
S = sum((h(k-j)*h(j))/(j+1) for j in (1..k-1))
return (h(k-1)/k-S)/(1+1/(k+1))
return numerator(h(2*n)*2^n*rising_factorial(1/2, n))
CROSSREFS
Cf. A097303 (see W. Lang link there for a similar numerator sequence which deviates for the first time at entry number 33. Expansion of GAMMA(z) in terms of 1/(k!*z^k) instead of 1/z^k).
Denominators of an asymptotic series for the factorial function (Stirling's formula with half-shift).
+10
6
1, 24, 1152, 414720, 39813120, 6688604160, 4815794995200, 115579079884800, 22191183337881600, 263631258054033408000, 88580102706155225088000, 27636992044320430227456000, 39797268543821419527536640000
COMMENTS
G_n = A182935(n)/ A144618(n). These rational numbers provide the coefficients for an asymptotic expansion of the factorial function. The relationship between these coefficients and the Bernoulli numbers are due to De Moivre, 1730 (see Laurie). (End)
Also denominators of polynomials mentioned in A144617. Also denominators of polynomials mentioned in A144622.
FORMULA
z! ~ sqrt(2 Pi) (z+1/2)^(z+1/2) e^(-z-1/2) Sum_{n>=0} G_n / (z+1/2)^n.
EXAMPLE
G_0 = 1, G_1 = -1/24, G_2 = 1/1152, G_3 = 1003/414720.
MAPLE
G := proc(n) option remember; local j, R;
R := seq(2*j, j=1..iquo(n+1, 2));
`if`(n=0, 1, add(bernoulli(j, 1/2)*G(n-j+1)/(n*j), j=R)) end:
MATHEMATICA
a[0] = 1; a[n_] := a[n] = Sum[ BernoulliB[j, 1/2]*a[n-j+1]/(n*j), {j, 2, n+1, 2}]; Table[a[n] // Denominator, {n, 0, 12}] (* Jean-François Alcover, Jul 26 2013, after Maple *)
AUTHOR
N. J. A. Sloane, Jan 15 2009, based on email from Chris Kormanyos (ckormanyos(AT)yahoo.com)
EXTENSIONS
Added more terms up to polynomial number u_12, v_12 for the denominators of u_k, v_k. Christopher Kormanyos (ckormanyos(AT)yahoo.com), Jan 31 2009
A-number in definition corrected - R. J. Mathar, Aug 05 2010 Edited and new definition by Peter Luschny, Feb 24 2011
Numerator of the coefficient of z^(-n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^2).
+10
6
1, -1, 1, 259193, -1036793, -201551328007, 9137074752049, 9142431862033871923, -11105299580705049589, -11003865617473929216508154207, 114467620015003245418244743007, 32505236416490926096399421788847363, -254505521478572052318535393350091231, -1828472168539763642032546635313363411876021
COMMENTS
1^(1^2)*2^(2^2)*...*n^(n^2) ~ A_2*n^(n^3/3+n^2/2+n/6)*exp(-n^3/9+n/12)*(Sum_{k>=0} b(k)/n^k)^n, where A_2 is the second Bendersky constant.
a(n) is the numerator of b(n).
FORMULA
Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence
c_0 = 1, c_n = (2/n) * Sum_{k=0..n-1} B_{n-k+3}*c_k/((n-j+1)*(n-k+2)*(n-k+3)) for n > 0.
a(n) is the numerator of c_n.
EXAMPLE
1^(1^2)*2^(2^2)*...*n^(n^2) ~ A_2*n^(n^3/3+n^2/2+n/6)*exp(-n^3/9+n/12)*(1 - 1/(360*n) + 1/(259200*n^2) + 259193/(1959552000*n^3) - 1036793/(2821754880000*n^4) - 201551328007/(5079158784000000*n^5) + ... ).
Denominator of the coefficient of z^(-n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^2).
+10
6
1, 360, 259200, 1959552000, 2821754880000, 5079158784000000, 76796880814080000000, 304115648023756800000000, 125121866615488512000000000, 258236518070374430146560000000000, 929651465053347948527616000000000000, 334674527419205261469941760000000000000, 920050700832433373350094438400000000000000
COMMENTS
1^(1^2)*2^(2^2)*...*n^(n^2) ~ A_2*n^(n^3/3+n^2/2+n/6)*exp(-n^3/9+n/12)*(Sum_{k>=0} b(k)/n^k)^n, where A_2 is the second Bendersky constant.
a(n) is the denominator of b(n).
FORMULA
Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence
c_0 = 1, c_n = (2/n) * Sum_{k=0..n-1} B_{n-k+3}*c_k/((n-j+1)*(n-k+2)*(n-k+3)) for n > 0.
a(n) is the denominator of c_n.
EXAMPLE
1^(1^2)*2^(2^2)*...*n^(n^2) ~ A_2*n^(n^3/3+n^2/2+n/6)*exp(-n^3/9+n/12)*(1 - 1/(360*n) + 1/(259200*n^2) + 259193/(1959552000*n^3) - 1036793/(2821754880000*n^4) - 201551328007/(5079158784000000*n^5) + ... ).
Numbers n such that n! has a square number of digits.
(Formerly M0830)
+10
5
0, 1, 2, 3, 7, 12, 18, 32, 59, 81, 105, 132, 228, 265, 284, 304, 367, 389, 435, 483, 508, 697, 726, 944, 1011, 1045, 1080, 1115, 1187, 1454, 1494, 1617, 1659, 1788, 1921, 2012, 2105, 2200, 2248, 2395, 2445, 2861, 2915, 3192, 3480, 3539, 3902, 3964, 4476
COMMENTS
Numbers whose square is represented by the number of digits of n!: 1, 2, 3, 4, 6, 9, 11, 13, 15, 21, 23, 24, 25, 28, 29, ..., . - Robert G. Wilson v, May 14 2014 In M. Gardner's book, see reference, there is a tree printout of 105! with 169 digits, where the bottom row consists of the 25 trailing zeros of 105!. M. Gardner does not explain if this is the only factorial that can be displayed in a similar tree form.
Proof: If m! has q^2 digits, hence the number of trailing zeros in m! must be equal to 2*q-1 to satisfy this triangular look; m = 105 satisfies these two conditions with q = 13 because 105! has 13^2 = 169 digits and 2*13-1 = 25 trailing zeros.
When m < 105 and m! has q^2 digits (m <= 81), then q <= 11 and the number of trailing zeros is <= 2*q - 3.
When m > 105 and m! has q^2 digits (m >= 132), then q >= 15 and the number of trailing zeros is >= 2*q + 2.
Hence, only 105! presents such a tree printout.
1
081
39675
8240290
900504101
30580032964
9720646107774
902579144176636
57322653190990515
3326984536526808240
339776398934872029657
99387290781343681609728
0000000000000000000000000
(End)
REFERENCES
M. Gardner, Mathematical Magic Show. Random House, NY, 1978, p. 55.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
MATHEMATICA
LogBase10Stirling[n_] := Floor[Log[10, 2 Pi n]/2 + n*Log[10, n/E] + Log[10, 1 + 1/(12n) + 1/(288n^2) - 139/(51840n^3) - 571/(2488320n^4) + 163879/(209018880n^5)]]; Select[ Range[ 4500], IntegerQ[ Sqrt[ (LogBase10Stirling[ # ] + 1)]] & ] (* The Mathematica coding comes from J. Stirling's expansion for the Gamma function; see the links. For more terms inside the last Log_10 function, use A001163 & A001164. Robert G. Wilson v, Apr 27 2014 *) Select[Range[0, 4500], IntegerQ[Sqrt[IntegerLength[#!]]]&] (* Harvey P. Dale, Sep 27 2018 *)
PROG
(PARI) isok(n) = issquare(#Str(n!)); \\ Michel Marcus, Sep 05 2015 (Magma) [k:k in [0..5000]| IsSquare(#Intseq(Factorial(k)))]; // Marius A. Burtea, Jan 04 2020
Numerators of rational coefficients related to Stirling's asymptotic series for the Gamma function.
+10
5
1, 2, 1, -4, 1, 8, -139, 16, -571, -8992, 163879, -334144, 5246819, 698752, -534703531, 23349012224, -4483131259, -1357305243136, 432261921612371, -6319924923392, 6232523202521089, 8773495082018816, -25834629665134204969, 49004477022654464, -1579029138854919086429
COMMENTS
The rational numbers SGGS = A264148/ A264149 (SGGS stands for 'Stirling Generalized Gamma Series') are a supersequence of the coefficients in Stirling's asymptotic series for the Gamma function A001163/ A001164 and of an asymptotic expansion of Ramanujan A090804/ A065973, further they appear in scaled form in an expansion of -W_{-1}(-e^{-1-x^2/2}) where W_{-1} is Lambert W function A005447/ A005446. Ramanujan's asymptotic expansion theta(n) = 1/3+4/(135n)-8/(2835n^2)- ... is considered in the literature also in the form 1-theta(n) (see for example formula (5) in the Choi link). It is this form to which we refer here.
FORMULA
Let SGGS(n) = h(n)*doublefactorial(n-1) where h(n) = 1 for n<=0 and for n>0 defined by the recurrence (h(k-1)/k - Sum_{j=1..k-1}((h(k-j)*h(j))/(j+1))/ (1+1/(k+1))) then a(n) = numerator(SGGS(n)).
MAPLE
h := proc(k) option remember; local j; `if`(k<=0, 1,
(h(k-1)/k-add((h(k-j)*h(j))/(j+1), j=1..k-1))/(1+1/(k+1))) end:
SGGS := n -> h(n)*doublefactorial(n-1):
MATHEMATICA
h[k_] := h[k] = If[k <= 0, 1, (h[k - 1]/k - Sum[h[k - j]*h[j]/(j + 1), {j, 1, k - 1}]) / (1 + 1/(k + 1))]; a[n_] := h[n]* Factorial2[n - 1] // Numerator; Table[a[n], {n, 0, 24}]
PROG
(Sage)
@cached_function
def h(k):
if k<=0: return 1
S = sum((h(k-j)*h(j))/(j+1) for j in (1..k-1))
return (h(k-1)/k-S)/(1+1/(k+1))
return numerator(h(n)*(n-1).multifactorial(2))
print([ A264148(n) for n in (0..17)])
CROSSREFS
A001163(n) = numerator(SGGS(2*n)) = numerator(SGGS(2*n)/2^(n+1)).
A001164(n) = denominator(SGGS(2*n)).
A090804(n) = numerator(SGGS(2*n+1)).
A065973(n) = denominator(SGGS(2*n+1)) = denominator(SGGS(2*n+1)/2^(n+1)).
A005447(n+1) = numerator(SGGS(n)/2^(n+1)).
A264150(n) = numerator(SGGS(2*n+1)/2^(n+1)).
Numerators of an asymptotic series for the Gamma function (even power series).
+10
5
1, -1, 19, -2561, 874831, -319094777, 47095708213409, -751163826506551, 281559662236405100437, -49061598325832137241324057, 5012066724315488368700829665081, -26602063280041700132088988446735433, 40762630349420684160007591156102493590477
COMMENTS
Let y = x+1/2 then Gamma(x+1) ~ sqrt(2*Pi)*((y/E)*Sum_{k>=0} r(k)/y^(2*k))^y as x -> oo and r(k) = A277000(k)/ A277001(k) (see example 6.1 in the Wang reference).
FORMULA
a(n) = numerator(b(2*n)) with b(n) = Y_{n}(0, z_2, z_3,..., z_n)/n! with z_k = k!*Bernoulli(k,1/2)/(k*(k-1)) and Y_{n} the complete Bell polynomials.
The rational numbers have the recurrence r(n) = (1/(2*n))*Sum_{m=0..n-1} Bernoulli(2*m+2,1/2)*r(n-m-1)/(2*m+1)) for n>=1, r(0)=1. - Peter Luschny, Sep 30 2016
EXAMPLE
The underlying rational sequence starts:
1, 0, -1/24, 0, 19/5760, 0, -2561/2903040, 0, 874831/1393459200, 0, ...
MAPLE
b := n -> CompleteBellB(n, 0, seq((k-2)!*bernoulli(k, 1/2), k=2..n))/n!:
# Alternatively the rational sequence by recurrence:
R := proc(n) option remember; local k; `if`(n=0, 1,
add(bernoulli(2*m+2, 1/2)* R(n-m-1)/(2*m+1), m=0..n-1)/(2*n)) end:
MATHEMATICA
CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}];
b[n_] := CompleteBellB[n, Join[{0}, Table[(k-2)! BernoulliB[k, 1/2], {k, 2, n}]]]/n!;
a[n_] := Numerator[b[2n]];
Number of decimal digits in (n!)!; A000197.
+10
3
1, 1, 1, 3, 24, 199, 1747, 16474, 168187, 1859934, 22228104, 286078171, 3949867548, 58284826485, 915905054360, 15276520209206, 269617872744249, 5021159048900643, 98417586560408168, 2025488254833817394, 43675043585825292775, 984729344827900257489, 23172929656443132617906
LINKS
Eric Weisstein's World of Mathematics, Factorial
MATHEMATICA
LogBase10Stirling[n_] := Floor[ Log[10, 2 Pi n]/2 + n*Log[10, n/E] + Log[10, 1 + 1/(12n) + 1/(288n^2) - 139/(51840n^3) - 571/(2488320n^4) + 163879/(209018880n^5) + 5246819/(75246796800n^6)]]; (* A001163/ A001164; good to at least a(1000) *) LogBase10Stirling[0] = LogBase10Stirling[1] = 0; Table[1 + LogBase10Stirling[n!], {n, 0, 101}] (* Robert G. Wilson v, Aug 05 2015 *)
PROG
(PARI) \\ Using 100 digits of precision.
a(n)=localprec(100); my(t=n!); return(floor((t*log(t)-t+1/2*log(2*Pi*t)+1/(12*t))/log(10)+1))\\ Robert Gerbicz, Jul 08 2008 (Magma) // Using about 100 more digits of precision than needed.
nMax:=30; SetDefaultRealField(RealField(Ceiling(Log(10, Factorial(nMax))+100))); a:=[]; for n in [0..nMax] do a[n+1]:=1+Floor(LogGamma(Factorial(n)+1)/Log(10)); end for; a; // Jon E. Schoenfield, Aug 07 2015
EXTENSIONS
A correspondent reported that terms a(17) - a(19) shown here were wrong. That's not true, they are correct. The correspondent was using Python, where the default precision was not large enough to calculate these terms correctly. Thanks to Brendan McKay, Max Alekseyev and Robert Gerbicz for confirming the entries. - N. J. A. Sloane, Jul 08 2008
Numerators of an asymptotic series for the Gamma function (G. Nemes)
+10
3
1, 0, 1, -1, -257, -53, 5741173, 37529, -710165119, -3376971533, 360182239526821, 104939254406053, -508096766056991140541, -70637580369737593, 289375690552473442964467, 796424971106808496421869
COMMENTS
G_n = A182912(n)/ A182913(n). These rational numbers provide the coefficients for an asymptotic expansion of the Gamma function.
REFERENCES
G. Nemes, More Accurate Approximations for the Gamma Function, Thai Journal of Mathematics Volume 9(1) (2011), 21-28.
FORMULA
Gamma(x+1) ~ x^x e^(-x) sqrt(2Pi(x+1/6)) Sum_{n>=0} G_n / (x+1/4)^n.
EXAMPLE
G_0 = 1, G_1 = 0, G_2 = 1/144, G_3 = -1/12960.
MAPLE
G := proc(n) option remember; local j, J;
J := proc(k) option remember; local j; `if`(k=0, 1,
(J(k-1)/k-add((J(k-j)*J(j))/(j+1), j=1..k-1))/(1+1/(k+1))) end:
add(J(2*j)*2^j*6^(j-n)*GAMMA(1/2+j)/(GAMMA(n-j+1)*GAMMA(1/2+j-n)), j=0..n)-add(G(j)*(-4)^(j-n)*(GAMMA(n)/(GAMMA(n-j+1)*GAMMA(j))), j=1..n-1) end:
MATHEMATICA
G[n_] := G[n] = Module[{j, J}, J[k_] := J[k] = Module[{j}, If[k == 0, 1, (J[k-1]/k - Sum[J[k-j]*J[j]/(j+1), {j, 1, k-1}])/(1+1/(k+1))]]; Sum[J[2*j]*2^j*6^(j-n)*Gamma[1/2+j]/(Gamma[n-j+1]*Gamma[1/2+j-n]), {j, 0, n}] - Sum[G[j]*(-4)^(j-n)*Gamma[n]/(Gamma[n-j+1]*Gamma[j]), {j, 1, n-1}]]; A182912[n_] := Numerator[G[n]]; Table[ A182912[i], {i, 0, 15}] (* Jean-François Alcover, Jan 06 2014, translated from Maple *)
Search completed in 0.021 seconds
|