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

0,6

Links

Table of n, a(n) for n=0..35.

Example

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]

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 *)

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).

Sequence in context: A030000 A367508 A221809 * A323836 A365893 A062520

Adjacent sequences: A291448 A291449 A291450 * A291452 A291453 A291454

Keyword

nonn,tabl

Author

Peter Luschny, Sep 07 2017

Status

approved