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

a(n) = 3^n*hypergeom([-n, -n - 1], [1], 1/3). - Detlef Meya, May 21 2024

From Peter Bala, Sep 02 2024: (Start)

The following formulas assume an offset of 1 (i.e., a(1) = 1, a(2) = 5, etc.):

a(n) = (1/3) * [x^n] ((2*x - 1)/(1 + x))^n = (1/3) * A255688(n).

a(n) = (1/3) * Sum_{k = 0..n} binomial(n, k)*binomial(n+k-1, k)*2^(n-k).

a(n) = (1/3) * 2^n * hypergeom([n, -n], [1], -1/2).

The Gauss congruences a(n*p^r) == a(n*p^(r-1)) (mod p^r) hold for all primes p >= 5 and all positive integers n and r. (End)

Maple

seq(simplify(3^n*hypergeom([-n, -n-1], [1], 1/3)), n = 0..20); # Peter Bala, Sep 02 2024

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 *)
a[n_] := 3^n*HypergeometricPFQ[{-n, -n - 1}, {1}, 1/3]; Flatten[Table[a[n], {n, 0, 21}]] (* Detlef Meya, May 21 2024 *)

Prog

(PARI) my(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.

Cf. A255688.

Sequence in context: A059273 A352175 A038744 * A265085 A158828 A264910

Adjacent sequences: A098660 A098661 A098662 * A098664 A098665 A098666

Keyword

easy,nonn,changed

Author

Paul Barry, Sep 20 2004

Status

approved