A262441 a(n) = Sum_{k=0..n+1}(binomial(n-1,k)/(k+1)*binomial(n+k+1,n-k)).

1, 2, 5, 16, 58, 226, 924, 3910, 16979, 75232, 338776, 1545886, 7132580, 33219086, 155963851, 737383488, 3507680650, 16776206680, 80622416976, 389123999656, 1885405316596, 9167409871040, 44717351734160, 218762640192838, 1073082055680180

Offset

0,2

Links

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

Formula

G.f.: 1/x-1/A(x), where A(x) is g.f. of A109081.

Recurrence: 2*(n+1)*(2*n - 1)*(19*n - 30)*a(n) = 20*(19*n^3 - 49*n^2 + 34*n - 6)*a(n-1) + 2*(n-2)*(38*n^2 - 79*n + 15)*a(n-2) + 3*(n-3)*(n-2)*(19*n - 11)*a(n-3). - Vaclav Kotesovec, Sep 23 2015

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

Mathematica

Join[{1}, Table[Sum[ Binomial[n-1, k] / (k+1) Binomial[ n+k+1, n-k], {k, 0, n+1}], {n, 25}]] (* Vincenzo Librandi, Sep 23 2015 *)

Prog

(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((1/x-1/A(x)), x, 0, 10);
(Magma) [&+[Binomial(n-1, k)/(k+1)*Binomial(n+k+1, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 23 2015
(PARI) a(n)=sum(k=0, n+1, (binomial(n-1, k)/(k+1)*binomial(n+k+1, n-k))) \\ Anders Hellström, Sep 23 2015

Crossrefs

Cf. A109081, A161798.

Sequence in context: A149978 A212263 A149979 * A328296 A300042 A019448

Adjacent sequences: A262438 A262439 A262440 * A262442 A262443 A262444

Keyword

nonn

Author

Vladimir Kruchinin, Sep 23 2015

Status

approved