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1, 4, 10, 25, 67, 172, 448, 1165, 3025, 7864, 20434, 53101, 137995, 358600, 931888, 2421673, 6293137, 16353820, 42498250, 110439121, 286995331, 745807444, 1938110800, 5036519125, 13088273857, 34012163632, 88386542578
COMMENTS
p(2,x) is row polynomial corresponding to triangle row A033842(2,m).
FORMULA
G.f.: x*(1+3*x+3*x^2)/(1-x-3*x^2-3*x^3)= x*p(2, x)/(1-x*p(2, x)) with x*p(2, x) G.f. for first column of A049324. Limit_{n-->oo} a(n+1)/a(n) = 3/(10^(1/3)-1) = 2.5986745... - Paul D. Hanna, Oct 13 2004
Triangle read by rows, the Bell transform of n!*binomial(2,n) (without column 0).
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1, 2, 1, 2, 6, 1, 0, 20, 12, 1, 0, 40, 80, 20, 1, 0, 40, 360, 220, 30, 1, 0, 0, 1120, 1680, 490, 42, 1, 0, 0, 2240, 9520, 5600, 952, 56, 1, 0, 0, 2240, 40320, 48720, 15120, 1680, 72, 1, 0, 0, 0, 123200, 332640, 184800, 35280, 2760, 90, 1, 0, 0, 0, 246400, 1786400
COMMENTS
Previous name was: A triangle of numbers related to triangle A049324. a(n,1) = A008279(2,n-1). a(n,m) =: S1(-2; n,m), a member of a sequence of lower triangular Jabotinsky matrices, including S1(1; n,m) = A008275 (signed Stirling first kind), S1(2; n,m) = A008297(n,m) (signed Lah numbers). a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))* A004747(n,m). The monic row polynomials E(n,x) := sum(a(n,m)*x^m,m=1..n), E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
FORMULA
a(n, m) = n!* A049324(n, m)/(m!*3^(n-m)); a(n, m) = (3*m-n+1)*a(n-1, m) + a(n-1, m-1), n >= m >= 1;
a(n, m) = 0, n<m; a(n, 0) = 0; a(1, 1) = 1.
E.g.f. for m-th column: ((x+x^2+(x^3)/3)^m)/m!.
a(n,m) = n!/(3^m * m!)*(Sum_{i=0..floor(m-n/3)} (-1)^i * binomial(m,i) * binomial(3*m-3*i,n)), 0 for empty sums. - Werner Schulte, Feb 20 2020
EXAMPLE
E.g. row polynomial E(3,x) = 2*x+6*x^2+x^3.
Triangle starts:
{1}
{2, 1}
{2, 6, 1}
{0, 20, 12, 1}
MATHEMATICA
rows = 11;
a[n_, m_] := BellY[n, m, Table[k! Binomial[2, k], {k, 0, rows}]];
PROG
(Sage) # uses[bell_matrix from A264428] # Adds 1, 0, 0, 0, ... as column 0 at the left side of the triangle.
bell_matrix(lambda n: factorial(n)*binomial(2, n), 8) # Peter Luschny, Jan 16 2016
A convolution triangle of numbers generalizing Pascal's triangle A007318.
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3
1, 6, 1, 16, 12, 1, 16, 68, 18, 1, 0, 224, 156, 24, 1, 0, 448, 840, 280, 30, 1, 0, 512, 3072, 2080, 440, 36, 1, 0, 256, 7872, 10896, 4160, 636, 42, 1, 0, 0, 14080, 42240, 28240, 7296, 868, 48, 1, 0, 0, 16896, 123904, 145376, 60720, 11704, 1136, 54, 1, 0, 0, 12288
FORMULA
a(n, m) = 4*(4*m-n+1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0; a(1, 1)=1. G.f. for m-th column: (x*p(3, x))^m, p(3, x) := 1+6*x+16*x^2+16*x^3 (row polynomial of A033842(3, m)).
EXAMPLE
{1}; {6,1}; {16,12,1}; {16,68,18,1}; {0,224,156,24,1}; ...
CROSSREFS
a(n, m) := s1(-3, n, m), a member of a sequence of triangles including s1(0, n, m)= A023531(n, m) (unit matrix) and s1(2, n, m)= A007318(n-1, m-1) (Pascal's triangle). s1(-1, n, m)= A030528, s1(-2, n, m)= A049324(n, m).
Triangle of coefficients of certain polynomials (exponents in increasing order), equivalent to A033842.
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1
1, 1, 1, 1, 3, 3, 1, 6, 16, 16, 1, 10, 50, 125, 125, 1, 15, 120, 540, 1296, 1296, 1, 21, 245, 1715, 7203, 16807, 16807, 1, 28, 448, 4480, 28672, 114688, 262144, 262144, 1, 36, 756, 10206, 91854, 551124, 2125764, 4782969, 4782969, 1, 45, 1200, 21000, 252000
COMMENTS
These polynomials p(n, x) appear in the W. Lang reference as c1(-(n+1);x), n >= 0 on p.12. The coefficients are given there in eq.(44) on p. 6. - Wolfdieter Lang, Nov 20 2015
FORMULA
a(n, m) = A033842(n, n-m) = binomial(n+1, m+1)*(n+1)^{m-1}, n >= m >= 0, else 0. a(3*n,2*n) = A001764(n)*(3*n+1)^(2*n); a(p*n,(p-1)*n) = binomial(p*n,n)/((p-1)*n+1)*(p*n+1)^((p-1)*n) for p > 0;
Sum_{m=0..n} (m+1)*a(n,m) = (n+2)^n;
Sum_{m=0..n} (-1)^m*(m+1)*a(n,m) = (-n)^n where 0^0 = 1;
p(n,x) = Sum_{m=0..n} a(n,m)*x^m = ((1+(n+1)*x)^(n+1)-1)/((n+1)^2*x).
(End)
EXAMPLE
The triangle a(n, m) begins:
n\m 0 1 2 3 4 5 6 7 ...
0: 1
1: 1 1
2: 1 3 3
3: 1 6 16 16
4: 1 10 50 125 125
5: 1 15 120 540 1296 1296
6: 1 21 245 1715 7203 16807 16807
7: 1 28 448 4480 28672 114688 262144 262144
... reformatted. - Wolfdieter Lang, Nov 20 2015
E.g. the third row {1,3,3} corresponds to polynomial p(2,x)= 1 + 3*x + 3*x^2.
MAPLE
seq(seq(binomial(n+1, m+1)*(n+1)^(m-1), m=0..n), n=0..10); # Robert Israel, Oct 19 2015
MATHEMATICA
Table[Binomial[n + 1, k + 1] (n + 1)^(k - 1), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Nov 19 2015 *)
PROG
(Magma) /* As triangle: */ [[Binomial(n+1, k+1)*(n+1)^(k-1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Nov 20 2015
CROSSREFS
a(n, 0)= A000012 (powers of 1), a(n, 1)= A000217 (triangular numbers), a(n, n)= A000272(n+1), n >= 0 (diagonal), a(n, n-1)= A000272(n+1), n >= 1.
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