A262442 a(n) = Sum_{k=0..n}(binomial(n-1,n-k)*binomial(n+k-1,n-k)).

1, 1, 3, 12, 53, 244, 1152, 5531, 26875, 131760, 650492, 3229368, 16105344, 80624935, 404913225, 2039146908, 10293657125, 52071019600, 263888886528, 1339540863092, 6809667715812, 34663102092960, 176655038497000, 901269559693104

Offset: 0

Author

Vladimir Kruchinin, Sep 23 2015

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A109081, A262440, A262441.

Programs

Magma

[&+[Binomial(n-1, n-k)*Binomial(n+k-1, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 23 2015

Mathematica

Join[{1}, Table[Sum[Binomial[n-1, n-k] Binomial[ n+k-1, n-k], {k, n}], {n, 25}]] (* Vincenzo Librandi, Sep 23 2015 *)

Maxima

a(n):=sum(binomial(n, k)*binomial(n+k-2, n-k-1), k, 0, n-1)/n;
A(x):=sum(a(n)*x^n, n, 1, 30);
taylor(x*diff(A(x),x)/A(x)-x^2*diff(1/x-1/A(x),x),x,0,10);

PARI

a(n) = sum(k=0,n,(binomial(n-1,n-k)*binomial(n+k-1,n-k))) \\ Anders Hellström, Sep 23 2015

Formula

G.f.: 1+A'(x)*(x*A(x)-x^2)/A(x)^2, where A(x) is g.f. of A109081.

Recurrence: 2*(n-1)*(2*n - 1)*(38*n^2 - 162*n + 163)*a(n) = 2*(380*n^4 - 2380*n^3 + 5200*n^2 - 4676*n + 1431)*a(n-1) + 2*(n-2)*(76*n^3 - 362*n^2 + 502*n - 189)*a(n-2) + 3*(n-3)*(n-2)*(38*n^2 - 86*n + 39)*a(n-3). - Vaclav Kotesovec, Sep 23 2015

a(n) = n*hypergeom([1 - n, 1 - n, n + 1], [1, 3/2], 1/4) for n >= 1. - Peter Luschny, Mar 07 2022