A136590 Triangle of trinomial logarithmic coefficients: A027907(n,k) = Sum_{i=0..k} T(k,i)*n^i/k!.
1, 0, 1, 0, 1, 1, 0, -4, 3, 1, 0, 6, -13, 6, 1, 0, 24, -10, -25, 10, 1, 0, -240, 394, -135, -35, 15, 1, 0, 720, -2016, 1834, -525, -35, 21, 1, 0, 5040, -5076, -3668, 5089, -1400, -14, 28, 1, 0, -80640, 170064, -110692, 14364, 9849, -3024, 42, 36, 1, 0, 362880, -1155024, 1339020, -672400, 118125, 12873, -5670, 150, 45, 1
Offset: 0
Examples
Triangle begins: 1; 0, 1; 0, 1, 1; 0, -4, 3, 1; 0, 6, -13, 6, 1; 0, 24, -10, -25, 10, 1; 0, -240, 394, -135, -35, 15, 1; 0, 720, -2016, 1834, -525, -35, 21, 1; 0, 5040, -5076, -3668, 5089, -1400, -14, 28, 1; 0, -80640, 170064, -110692, 14364, 9849, -3024, 42, 36, 1; 0, 362880, -1155024, 1339020, -672400, 118125, 12873, -5670, 150, 45, 1; ... Trinomial coefficients can be calculated as illustrated by: A027907(4,3) = (T(3,0)*4^0 + T(3,1)*4^1 + T(3,2)*4^2 + T(3,3)*4^3)/3! = (0 - 4*4 + 3*4^2 + 1*4^3)/3! = 96/6 = 16.
Crossrefs
Programs
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Maple
# The function BellMatrix is defined in A264428. BellMatrix(n -> n!*(modp(n+1,3)-modp(n,3)), 9); # Peter Luschny, Jan 27 2016
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Mathematica
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]]; rows = 11; M = BellMatrix[#!*(Mod[# + 1, 3] - Mod[#, 3])&, rows]; Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *) -
PARI
{T(n,k)=n!/k!*polcoeff(log(1+x+x^2 +x*O(x^n))^k,n)} for(n=0,10, for(k=0,n, print1( T(n,k),", "));print("")) -
Sage
# uses[bell_matrix from A264428] bell_matrix(lambda n: A136591(n+1), 10) # Peter Luschny, Jan 18 2016
Formula
E.g.f. of column k = log(1 + x + x^2)^k / k! for k>=0.
Central trinomial coefficients: A002426(n) = Sum_{k=0..n} T(n,k)*n^k/n!.
Comments