A291975 - OEIS

%I #14 Jan 21 2020 18:26:19

%S 1,1,36,6271,3086331,3309362716,6626013560301,22360251390209461,

%T 118214069460929849196,926848347928901638652131,

%U 10326354052861964007954596391,157987763647812764532709527137476,3227443522308474152275617569919520761

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

%C Row sums of A291452.

%H Andrew Howroyd, <a href="/A291975/b291975.txt">Table of n, a(n) for n = 0..100</a>

%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(4*n-1,4*k-1) * a(n-k). - _Ilya Gutkovskiy_, Jan 21 2020

%p A291975 := proc(n) exp((cos(z) + cosh(z))/2 - 1):

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

%p seq(A291975(n), n=0..12);

%t P[m_, n_] := P[m, n] = If[n == 0, 1, Sum[Binomial[m*n, m*k]*P[m, n - k]*x, {k, 1, n}]];

%t a[n_] := Module[{cl = CoefficientList[P[4, n], x]}, Sum[cl[[k + 1]]/k!, {k, 0, n}]];

%t Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Jul 23 2019, after _Peter Luschny_ in A291452 *)

%o (PARI) seq(n)={my(a=vector(n+1)); a[1]=1; for(n=1, n, a[1+n]=sum(k=1, n, binomial(4*n-1, 4*k-1) * a[1+n-k])); a} \\ _Andrew Howroyd_, Jan 21 2020

%Y Cf. A291452.

%K nonn

%O 0,3

%A _Peter Luschny_, Sep 07 2017