A352526 a(n) = Product_{k=0..n} Nimsum (2*k + 2), with Nimsum (2 + 2) = 0 replaced by 1.

2, 2, 12, 48, 480, 3840, 53760, 645120, 11612160, 185794560, 4087480320, 81749606400, 2125489766400, 51011754393600, 1530352631808000, 42849873690624000, 1456895705481216000, 46620662575398912000, 1771585177865158656000, 63777066403145711616000, 2678636788932119887872000

Offset: 0

Author

Peter McNair, Mar 19 2022

Keywords

• nonn

Comments

Nimsum 2*k + 2 = A004443(2*k).

Sum_{n>0} 1/a(n) = 1/sqrt(e) = A092605.

Sum_{n>0} 1/a(2*n-1) = sinh(1/2) = A334367.

Sum_{n>0} 1/a(2*n) = cosh(1/2) - 2*sinh(1/2).

a(n)/2^n = abs(A265376(n+1)) = Product_{k=0..n} Nimsum k + 1, with Nimsum 1 + 1 = 0 replaced by 1, n > 0.

Crossrefs

Cf. A004443, A092605, A265376, A334367.

Programs

Mathematica

a[n_] := Product[If[k == 1, 1, BitXor[2*k, 2]], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Mar 19 2022 *)

PARI

a(n) = 2*prod(k=2,n,bitxor(2*k, 2))

Formula

a(n) = 2*Product_{k=2..n} A004443(2*k).

a(n) = 2^(n-1)*(n+1)!/floor((n+1)/2), n > 0.

a(n) = 2^(n-1)*(1+(-1)^n)*((n-1)!+n!)-((-1)^n-1)*(2*n)!!/2, n > 0.

a(n) = 2*a(n-1)*(n+(-1)^n), n > 1, with a(1) = 2.