A291451 Triangle read by rows, expansion of e.g.f. exp(x*(exp(z)/3 + 2*exp(-z/2)* cos(z*sqrt(3)/2)/3 - 1)), nonzero coefficients of z.

1, 0, 1, 0, 1, 10, 0, 1, 84, 280, 0, 1, 682, 9240, 15400, 0, 1, 5460, 260260, 1401400, 1401400, 0, 1, 43690, 7128576, 99379280, 285885600, 190590400, 0, 1, 349524, 193360720, 6600492080, 42549306800, 76045569600, 36212176000

Offset: 0

Author

Peter Luschny, Sep 07 2017

Keywords

• nonn
• tabl

Examples

			Triangle starts:
[1]
[0, 1]
[0, 1,    10]
[0, 1,    84,     280]
[0, 1,   682,    9240,    15400]
[0, 1,  5460,  260260,  1401400,   1401400]
[0, 1, 43690, 7128576, 99379280, 285885600, 190590400]
		

Crossrefs

Cf. A048993 (m=1), A156289 (m=2), this seq. (m=3), A291452 (m=4).

Diagonal: A000012 (m=1), A001147 (m=2), A025035 (m=3), A025036 (m=4).

Row sums: A000110 (m=1), A005046 (m=2), A291973 (m=3), A291975 (m=4).

Alternating row sums: A000587 (m=1), A260884 (m=2), A291974 (m=3), A291976 (m=4).

Programs

Maple

CL := (f, x) -> PolynomialTools:-CoefficientList(f, x):
A291451_row := proc(n) exp(x*(exp(z)/3+2*exp(-z/2)*cos(z*sqrt(3)/2)/3-1)):
series(%, z, 66): CL((3*n)!*coeff(series(%,z,3*(n+1)),z,3*n),x) end:
for n from 0 to 7 do A291451_row(n) od;
# Alternative:
A291451row := proc(n) local P; P := proc(m, n) option remember;
if n = 0 then 1 else add(binomial(m*n, m*k)*P(m, n-k)*x, k=1..n) fi end:
CL(P(3, n), x); seq(%[k+1]/k!, k=0..n) end: # Peter Luschny, Sep 03 2018

Mathematica

P[m_, n_] := P[m, n] = If[n == 0, 1, Sum[Binomial[m*n, m*k]*P[m, n - k]*x, {k, 1, n}]];
row[n_] := Module[{cl = CoefficientList[P[3, n], x]}, Table[cl[[k + 1]]/k!, {k, 0, n}]];
Table[row[n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Jul 23 2019, after Peter Luschny *)