A098663 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+1,k+1) * 3^k.

1, 5, 30, 193, 1286, 8754, 60460, 421985, 2968902, 21019510, 149572292, 1068795930, 7664092060, 55121602436, 397464604440, 2872406652001, 20799171328070, 150869330458830, 1096046132412628, 7973709600124958, 58081342410990516, 423551998861478140

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

G.f.: ((1+2*x) - sqrt(1-8*x+4*x^2))/(6*x*sqrt(1-8*x+4*x^2)).

E.g.f.: exp(4x)*(BesselI(0, 2*sqrt(3)*x) + BesselI(1, 2*sqrt(3)*x)/sqrt(3)).

Recurrence: (n+1)*(2*n-1)*a(n) = 2*(8*n^2-3)*a(n-1) - 4*(n-1)*(2*n+1)*a(n-2). - Vaclav Kotesovec, Oct 15 2012

a(n) ~ sqrt(12+7*sqrt(3))*(4+2*sqrt(3))^n/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 15 2012

Mathematica

Table[Sum[Binomial[n, k]Binomial[n+1, k+1]3^k, {k, 0, n}], {n, 0, 20}] (* Harvey P. Dale, Nov 08 2011 *)

Prog

(PARI) x='x+O('x^66); Vec(((1+2*x)-sqrt(1-8*x+4*x^2))/(6*x*sqrt(1-8*x+4*x^2))) \\ Joerg Arndt, May 12 2013

Crossrefs

Fourth binomial transform of A098662.

Sequence in context: A059273 A352175 A038744 * A265085 A158828 A264910

Adjacent sequences: A098660 A098661 A098662 * A098664 A098665 A098666

Keyword

easy,nonn

Author

Paul Barry, Sep 20 2004

Status

approved