A138415 Binomial transform of A000957.

0, 1, 2, 4, 10, 31, 110, 421, 1686, 6961, 29392, 126292, 550360, 2426503, 10803802, 48507844, 219377950, 998436793, 4569488372, 21016589074, 97090411020, 450314942683, 2096122733212, 9788916220519, 45850711498860, 215348942668681, 1013979873542690, 4785437476592806

Offset: 0

Author

N. J. A. Sloane, May 08 2008

Keywords

• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000957, A138461.

Programs

Mathematica

Table[Sum[Binomial[n, k]*(2^k * (2*k-1)!! * Hypergeometric2F1Regularized[2, 2*k+1, k+2, -1] - 3*(-1)^k/2^(k+1)), {k, 1, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 30 2017 *)
RecurrenceTable[{a[0]==0,a[1]==1,a[2]==2,a[3]==4,a[n]==(3(5n-6)a[n-1]-(29n-57) a[n-2]+3(7n-18)a[n-3]-5(n-3)a[n-4])/(2n)},a,{n,30}] (* Harvey P. Dale, Nov 22 2022 *)

Formula

From Vaclav Kotesovec, Oct 30 2017: (Start)

Recurrence: 2*n*a(n) = 3*(5*n - 6)*a(n-1) - (29*n - 57)*a(n-2) + 3*(7*n - 18)*a(n-3) - 5*(n-3)*a(n-4).

a(n) ~ 5^(n + 3/2) / (72 * sqrt(Pi) * n^(3/2)). (End)