A291974 - OEIS

%I #11 Jan 27 2023 09:03:05

%S 1,-1,9,-197,6841,-254801,-3000807,3691567683,-717149457463,

%T -3166484321001,70729161470807849,-27375562310313650357,

%U -6307300288015827588199,14726712291264935798753279,-4956785715421801286491780487,-9984523503726123391084330853037

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

%C Alternating row sums of A291451.

%H Alois P. Heinz, <a href="/A291974/b291974.txt">Table of n, a(n) for n = 0..205</a>

%p A291974 := proc(n) exp(-(exp(z)/3+2*exp(-z/2)*cos(z*sqrt(3)/2)/3-1)):

%p (3*n)!*coeff(series(%, z, 3*(n+1)), z, 3*n) end:

%p seq(A291974(n), n=0..15);

%p # second Maple program:

%p b:= proc(n, t) option remember; `if`(n=0, 1-2*t, add(

%p       b(n-3*j, 1-t)*binomial(n-1, 3*j-1), j=1..n/3))

%p     end:

%p a:= n-> b(3*n, 0):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Aug 14 2019

%t 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 a[n_] := b[3n, 0];

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jan 27 2023, after _Alois P. Heinz_ *)

%Y Cf. A291451.

%K sign

%O 0,3

%A _Peter Luschny_, Sep 07 2017