A144635 a(n) = 5^n*Sum_{ k=0..n } binomial(2*k,k)/5^k.

1, 7, 41, 225, 1195, 6227, 32059, 163727, 831505, 4206145, 21215481, 106782837, 536618341, 2693492305, 13507578125, 67693008145, 339066121115, 1697664211795, 8497396194275, 42522326235175, 212749477704695, 1064285646397915, 5323532330953295, 26625895085494075

Offset: 0

Author

N. J. A. Sloane, Jan 21 2009

Keywords

• nonn

Crossrefs

Cf. A000032, A006134, A082590, A132310, A002457.

Programs

Mathematica

Table[5^n Sum[Binomial[2k,k]/5^k,{k,0,n}],{n,0,30}] (* Harvey P. Dale, Aug 08 2011 *)
Round@Table[5^(n + 1/2) - 2^(n + 1) (2 n + 1)!! Hypergeometric2F1[1, n + 3/2, n + 2, 4/5]/(5 (n + 1)!), {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster - Vladimir Reshetnikov, Oct 14 2016 *)

PARI

a(n) = 5^n*sum(k=0, n, binomial(2*k,k)/5^k); \\ Michel Marcus, Oct 14 2016

Formula

From Vaclav Kotesovec, Jun 12 2013: (Start)

G.f.: 1/((1-5*x)*sqrt(1-4*x)).

Recurrence: n*a(n) = (9*n-2)*a(n-1) - 10*(2*n-1)*a(n-2).

a(n) ~ 5^(n+1/2). (End)

a(n) = 5^(n+1/2) - 2^(n+1)*(2*n+1)!!*hypergeom([1,n+3/2], [n+2], 4/5)/(5*(n+1)!). - Vladimir Reshetnikov, Oct 14 2016

a(n) = Sum_{k=0..n} C(2*n+1,n-k)*A000032(2*k+1). - Vladimir Kruchinin Jan 14 2025