A291976 - OEIS

%I #11 Jan 27 2023 09:04:30

%S 1,-1,34,-5281,2185429,-1854147586,2755045819549,-6440372006517541,

%T 21861211462545555394,-100916681831006840635021,

%U 596756926975162013357972089,-4237398636260867429185819175026,32919774165127854788267224335178009

%N a(n) = (4*n)! * [z^(4*n)] exp(1 - (cos(z) + cosh(z))/2).

%C Alternating row sums of A291452.

%H Alois P. Heinz, <a href="/A291976/b291976.txt">Table of n, a(n) for n = 0..155</a>

%p A291976 := proc(n) exp(1 - (cos(z) + cosh(z))/2):

%p (4*n)!*coeff(series(%, z, 4*(n+1)), z, 4*n) end:

%p seq(A291976(n), n=0..12);

%p # second Maple program:

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

%p       b(n-4*j, 1-t)*binomial(n-1, 4*j-1), j=1..n/4))

%p     end:

%p a:= n-> b(4*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-4j, 1-t] * Binomial[n-1, 4j-1], {j, 1, n/4}]];

%t a[n_] := b[4n, 0];

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

%Y Cf. A291452.

%K sign

%O 0,3

%A _Peter Luschny_, Sep 07 2017