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

Author

Paul D. Hanna, Jan 10 2008

Keywords

• sign
• tabl

Comments

A027907 is the triangle of trinomial coefficients.

The Bell transform of A136591(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

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

Cf. columns: A136591, A136592, A136593; A136594 (unsigned row sums); A136595 (matrix inverse); A027907, A002426.

Programs

Maple

# The function BellMatrix is defined in A264428.
BellMatrix(n -> n!*(modp(n+1,3)-modp(n,3)), 9); # Peter Luschny, Jan 27 2016

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!.