A291452 Triangle read by rows, expansion of e.g.f. exp(x*(cos(z) + cosh(z) - 2)/2), nonzero coefficients of z.

1, 0, 1, 0, 1, 35, 0, 1, 495, 5775, 0, 1, 8255, 450450, 2627625, 0, 1, 130815, 35586525, 727476750, 2546168625, 0, 1, 2098175, 2941884000, 181262956875, 1932541986375, 4509264634875

Offset

0,6

Links

Table of n, a(n) for n=0..27.

Example

Triangle starts:
[1]
[0, 1]
[0, 1,      35]
[0, 1,     495,       5775]
[0, 1,    8255,     450450,      2627625]
[0, 1,  130815,   35586525,    727476750,    2546168625]
[0, 1, 2098175, 2941884000, 181262956875, 1932541986375, 4509264634875]

Maple

CL := (f, x) -> PolynomialTools:-CoefficientList(f, x):
A291452_row := proc(n) exp(x*(cos(z)+cosh(z)-2)/2):
series(%, z, 88): CL((4*n)!*coeff(series(%, z, 4*(n+1)), z, 4*n), x) end:
for n from 0 to 7 do A291452_row(n) od;
# Alternative:
A291452row := 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(4, 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[4, n], x]}, Table[cl[[k + 1]]/k!, {k, 0, n}]];
Table[row[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Jul 23 2019, after Peter Luschny *)

Crossrefs

Cf. A048993 (m=1), A156289 (m=2), A291451 (m=3), this seq. (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: A236237 A067156 A365894 * A343981 A174593 A104785

Adjacent sequences: A291449 A291450 A291451 * A291453 A291454 A291455

Keyword

nonn,tabl

Author

Peter Luschny, Sep 07 2017

Status

approved