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

%I A291974 #11 Jan 27 2023 09:03:05
%S A291974 1,-1,9,-197,6841,-254801,-3000807,3691567683,-717149457463,
%T A291974 -3166484321001,70729161470807849,-27375562310313650357,
%U A291974 -6307300288015827588199,14726712291264935798753279,-4956785715421801286491780487,-9984523503726123391084330853037
%N A291974 a(n) = (3*n)! * [z^(3*n)] exp(-(exp(z)/3 + 2*exp(-z/2)*cos(z*sqrt(3)/2)/3 - 1)).
%C A291974 Alternating row sums of A291451.
%H A291974 Alois P. Heinz, <a href="/A291974/b291974.txt">Table of n, a(n) for n = 0..205</a>
%p A291974 A291974 := proc(n) exp(-(exp(z)/3+2*exp(-z/2)*cos(z*sqrt(3)/2)/3-1)):
%p A291974 (3*n)!*coeff(series(%, z, 3*(n+1)), z, 3*n) end:
%p A291974 seq(A291974(n), n=0..15);
%p A291974 # second Maple program:
%p A291974 b:= proc(n, t) option remember; `if`(n=0, 1-2*t, add(
%p A291974       b(n-3*j, 1-t)*binomial(n-1, 3*j-1), j=1..n/3))
%p A291974     end:
%p A291974 a:= n-> b(3*n, 0):
%p A291974 seq(a(n), n=0..20);  # _Alois P. Heinz_, Aug 14 2019
%t A291974 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}]];
%t A291974 a[n_] := b[3n, 0];
%t A291974 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jan 27 2023, after _Alois P. Heinz_ *)
%Y A291974 Cf. A291451.
%K A291974 sign
%O A291974 0,3
%A A291974 _Peter Luschny_, Sep 07 2017