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Number of labeled rooted trees of nonempty sets with n points. (Each node is a set of 1 or more points.)
+10
20
1, 3, 16, 133, 1521, 22184, 393681, 8233803, 198342718, 5408091155, 164658043397, 5537255169582, 203840528337291, 8153112960102283, 352079321494938344, 16325961781591781401, 809073412162081974237, 42674870241038732398720, 2386963662244981472850709
FORMULA
E.g.f.: B(exp(x)-1) where B is e.g.f. of A000169. E.g.f.: Series_Reversion( log(1 + x*exp(-x)) ). - Paul D. Hanna, Jan 24 2016 a(n) = Sum_{k=1..n} Stirling2(n, k)*k^(k-1). - Vladeta Jovovic, Sep 17 2003 a(n) ~ sqrt(1+exp(1)) * n^(n-1) / (exp(n) * (log(1+exp(-1)))^(n-1/2)). - Vaclav Kotesovec, Feb 17 2014
EXAMPLE
G.f. = x + 3*x^2 + 16*x^3 + 133*x^4 + 1521*x^5 + 22184*x^6 + 393681*x^7 + ...
MATHEMATICA
nn=20; t=Sum[n^(n-1)x^n/n!, {n, 1, nn}]; Range[0, nn]!CoefficientList[ ComposeSeries[ Series[t, {x, 0, nn}], Series[Exp[x]-1 , {x, 0, nn}]], x] (* Geoffrey Critzer, Sep 16 2012 *)
PROG
(PARI) {a(n) = sum( k=1, n, stirling(n, k, 2) * k^(k - 1))}; /* Michael Somos, Jun 09 2012 */ (PARI) {a(n) = n! * polcoeff( serreverse( log(1 + x*exp(-x +x*O(x^n))) ), n)}
E.g.f.: 1/(1 + LambertW(1-exp(x))), where LambertW() is the Lambert W-function.
+10
12
1, 1, 5, 40, 447, 6421, 112726, 2338799, 55990213, 1519122598, 46066158817, 1543974969769, 56677405835276, 2261488166321697, 97455090037460785, 4510770674565054000, 223183550978156866507, 11755122645815049275521, 656670295411196201190366, 38779502115371642484125915, 2413908564514961126280655257
LINKS
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
FORMULA
a(0) = 1, a(n) = Sum_{k=1..n} Stirling2(n,k)*k^k.
a(n) ~ n^n / (sqrt(1+exp(1)) * (log(1+exp(-1)))^(n+1/2) * exp(n)). - Vaclav Kotesovec, Feb 17 2017
EXAMPLE
E.g.f.: A(x) = 1 + x/1! + 5*x^2/2! + 40*x^3/3! + 447*x^4/4! + 6421*x^5/5! + 112726*x^6/6! + ...
MAPLE
b:= proc(n, m) option remember;
`if`(n=0, m^m, m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
MATHEMATICA
Range[0, 20]! CoefficientList[Series[1/(1 + ProductLog[1 - Exp[x]]), {x, 0, 20}], x]
Join[{1}, Table[Sum[StirlingS2[n, k] k^k, {k, 1, n}], {n, 1, 20}]]
PROG
(PARI) x='x+O('x^50); Vec(serlaplace(1/(1 + lambertw(1-exp(x))))) \\ G. C. Greubel, Nov 12 2017
Number of labeled rooted connected graphs where every block is a complete graph.
+10
9
0, 1, 2, 12, 116, 1555, 26682, 558215, 13781448, 392209380, 12641850510, 455198725025, 18109373455164, 788854833679549, 37343190699472322, 1908871649888004240, 104789417805394595600, 6148562290130009617619
COMMENTS
Equivalently, rooted labeled spanning trees in the complete hypergraph on n vertices (all hyperedges having cardinality 2 or greater).
REFERENCES
Warren D. Smith and David Warme, Paper in preparation, 2002.
FORMULA
Recurrence: a(1) = 1, a(n) = Sum_{k=1}^{n-1} Bell(k) / k! Sum_{a_j > 0, Sum_{j=1}^k a_j = n-1} {{n-1} choose {a_1, a_2, ..., a_k }} \prod_{j=1}^k a(a_j) for n > 1, where Bell(k) = A000110(k). - Warren D. Smith, Feb 23 1998 a(n) = Sum_{i=0...n-1} S(n-1, i) n^i, where S(N, M) are Stirling numbers of the second kind - David Warme, Mar 25 1998
E.g.f. satisfies A(x)=x*exp(exp(A(x))-1).
Let X_{mu} be a Poisson random variable with mean mu: P(X_{mu} = K) = e^{-mu} mu^K / K!. The n-th moment of X_{mu} is E[X_{mu}^n] = sum_{i=0}^n S(n, i) mu^i. Therefore a(n) = E[X_n^{n-1}]. - Langworth Withers, May 25 2000
Dobinski-type formula: a(n) = 1/e^n*sum {k = 0..inf} n^k*k^(n-1)/k!. Cf. A030019 and A052888. For a refinement of this sequence see A210586. - Peter Bala, Apr 05 2012 a(n) ~ exp((1/LambertW(1)-2)*n) * n^(n-1) / (sqrt(1+LambertW(1)) * LambertW(1)^(n-1)). - Vaclav Kotesovec, Jan 22 2014
MATHEMATICA
f[n_] := Sum[ n^i*StirlingS2[n - 1, i], {i, 0, n - 1}]; Array[f, 18, 0] (* Robert G. Wilson v, Apr 05 2012 *) Table[If[n == 0, 0, BellB[n - 1, n]], {n, 0, 100}] (* Emanuele Munarini, May 23 2014 *)
PROG
(Maxima) a(n):=if n=0 then 0 else sum(stirling2(n-1, k)*n^k, k, 0, n);
(PARI) for(n=0, 30, print1(sum(k=0, n-1, stirling(n-1, k, 2)*n^k), ", ")) \\ G. C. Greubel, Nov 17 2017
Number of trees of nonempty sets with n points. (Each node is a set of 1 or more points.)
+10
6
1, 1, 2, 3, 7, 14, 35, 85, 231, 633, 1845, 5461, 16707, 51945, 164695, 529077, 1722279, 5664794, 18813369, 62996850, 212533226, 721792761, 2466135375, 8471967938, 29249059293, 101440962296, 353289339927, 1235154230060, 4333718587353, 15255879756033
COMMENTS
Also the number of non-isomorphic connected multigraphs with loops with n edges and multiset density -1, where the multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices. - Gus Wiseman, Nov 28 2018
FORMULA
G.f.: B(x) - B^2(x)/2 + B(x^2)/2, where B(x) is g.f. for A036249.
MATHEMATICA
max = 30; B[_] = 1; Do[B[x_] = x*Exp[Sum[(B[x^k] + x^k)/k + O[x]^n, {k, 1, n}]] // Normal, {n, 1, max}]; A[x_] = B[x] - B[x]^2/2 + B[x^2]/2; CoefficientList[1 + A[x] + O[x]^max, x] (* Jean-François Alcover, Jan 28 2019 *)
CROSSREFS
Cf. A000055, A007718, A007719, A038052, A191646, A303837, A321155, A321229, A321254, A321256, A322111.
G.f.: B(x/(1-x)) where B is g.f. of A000169.
+10
4
1, 3, 14, 98, 944, 11642, 175108, 3108310, 63601168, 1473864722, 38152990484, 1091172974102, 34169139856024, 1162736848398010, 42723615842296540, 1685853467536076798, 71101435046807892512, 3191843270961299033762, 151956292916451992949028
FORMULA
E.g.f.: int(exp(x)*(-LambertW(-x)/(1+LambertW(-x))/x), x). a(n) = Sum_{k=0..n-1} binomial(n-1, k)*(k+1)^k. - Vladeta Jovovic, Apr 12 2003
MATHEMATICA
CoefficientList[Series[E^x*(-LambertW[-x]/(1+LambertW[-x])/x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Feb 17 2014 *)
Expansion of e.g.f. 1 - LambertW(-log(1+x))*(2 + LambertW(-log(1+x)))/2.
+10
1
1, 1, 0, 2, 3, 44, 260, 3534, 40796, 658440, 11066184, 220005840, 4750650432, 114430365048, 2993377996440, 85208541290040, 2611784941760640, 85941161628865344, 3018822193183216320, 112805065528683216192, 4467115744449046110720, 186900232401341222964480, 8237944325702047624948224
FORMULA
a(n) = Sum_{k=0..n} Stirling1(n,k)* A000272(k). a(n) ~ n^(n-2) / ((exp(exp(-1))-1)^(n - 3/2) * exp(n - 3*(1 - exp(-1))/2)). - Vaclav Kotesovec, Jan 20 2019
MAPLE
seq(n!*coeff(series(1-LambertW(-log(1+x))*(2+LambertW(-log(1+x)))/2, x=0, 23), x, n), n=0..22); # Paolo P. Lava, Jan 28 2019
MATHEMATICA
nmax = 22; CoefficientList[Series[1 - LambertW[-Log[1 + x]] (2 + LambertW[-Log[1 + x]])/2, {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[StirlingS1[n, k] k^(k - 2), {k, n}], {n, 22}]]
PROG
(PARI) {a(n) = if(n==0, 1, sum(k=1, n, stirling(n, k, 1)*k^(k-2)))};
(Magma) [n le 0 select 1 else (&+[StirlingFirst(n, k)*k^(k-2): k in [1..n]]): n in [0..25]]; // G. C. Greubel, Feb 07 2019 (Sage) [1] + [sum((-1)^(k+n)*stirling_number1(n, k)*k^(k-2) for k in (1..n)) for n in (1..25)] # G. C. Greubel, Feb 07 2019
Triangle read by rows: T(n,k) is the number of labeled quasi-loop-threshold graphs on vertex set [n] with k components, for n >= 1 and 1 <= k <= n.
+10
0
2, 3, 4, 16, 18, 8, 133, 155, 72, 16, 1521, 1810, 910, 240, 32, 22184, 26797, 14145, 4180, 720, 64, 393681, 480879, 262514, 83230, 16520, 2016, 128, 8233803, 10144283, 5675866, 1888873, 409360, 58912, 5376, 256
COMMENTS
The family of quasi-loop-threshold graphs is the smallest family of looped graphs that contains K_1 (a single vertex) and K^loop_1 (a single looped vertex), and is closed under taking unions and adding looped dominating vertices (looped, and adjacent to everything previously added).
FORMULA
See Section 1.4 of Galvin, Wesley and Zacovic link for two methods to compute T(n,k).
EXAMPLE
Triangle begins:
2;
3, 4;
16, 18, 8;
133, 155, 72, 16;
1521, 1810, 910, 240, 32;
22184, 26797, 14145, 4180, 720, 64;
393681, 480879, 262514, 83230, 16520, 2016, 128;
8233803, 10144283, 5675866, 1888873, 409360, 58912, 5376, 256;
...
MATHEMATICA
qltconn[0] = 0; qltconn[1] = 2; qltconn[n_] := qltconn[n] = Sum[StirlingS2[n, k]*(k^(k - 1)), {k, 1, n}] (*qltconn is the number of connected quasi loop threshold graphs on n vertices*); T[n_, l_] := T[n, l] := (Factorial[n]/Factorial[l])*Coefficient[(Sum[(qltconn[k]*(x^k))/Factorial[k], {k, 1, n}])^l, x, n]; Table[T[n, l], {n, 1, 12}, {l, 1, n}]
CROSSREFS
Except at n=1, the first column is A048802 ( A048802 takes value 1 at n=1).
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