A291974 a(n) = (3*n)! * [z^(3*n)] exp(-(exp(z)/3 + 2*exp(-z/2)*cos(z*sqrt(3)/2)/3 - 1)).

1, -1, 9, -197, 6841, -254801, -3000807, 3691567683, -717149457463, -3166484321001, 70729161470807849, -27375562310313650357, -6307300288015827588199, 14726712291264935798753279, -4956785715421801286491780487, -9984523503726123391084330853037

Offset: 0

Author

Peter Luschny, Sep 07 2017

Keywords

• sign

Comments

Alternating row sums of A291451.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..205

Crossrefs

Cf. A291451.

Programs

Maple

A291974 := proc(n) exp(-(exp(z)/3+2*exp(-z/2)*cos(z*sqrt(3)/2)/3-1)):
(3*n)!*coeff(series(%, z, 3*(n+1)), z, 3*n) end:
seq(A291974(n), n=0..15);
# second Maple program:
b:= proc(n, t) option remember; `if`(n=0, 1-2*t, add(
      b(n-3*j, 1-t)*binomial(n-1, 3*j-1), j=1..n/3))
    end:
a:= n-> b(3*n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 14 2019

Mathematica

b[n_, t_] := b[n, t] = If[n == 0, 1-2t, Sum[b[n-3j, 1-t] * Binomial[n-1, 3j-1], {j, 1, n/3}]];
a[n_] := b[3n, 0];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 27 2023, after Alois P. Heinz *)