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A098663 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+1,k+1) * 3^k. 2
1, 5, 30, 193, 1286, 8754, 60460, 421985, 2968902, 21019510, 149572292, 1068795930, 7664092060, 55121602436, 397464604440, 2872406652001, 20799171328070, 150869330458830, 1096046132412628, 7973709600124958, 58081342410990516, 423551998861478140 (list; graph; refs; listen; history; text; internal format)
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: A006773 A059273 A038744 * A265085 A158828 A264910

Adjacent sequences:  A098660 A098661 A098662 * A098664 A098665 A098666

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Sep 20 2004

STATUS

approved

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Last modified July 11 20:03 EDT 2020. Contains 335652 sequences. (Running on oeis4.)