A262441 - OEIS

%I #23 Sep 08 2022 08:46:14

%S 1,2,5,16,58,226,924,3910,16979,75232,338776,1545886,7132580,33219086,

%T 155963851,737383488,3507680650,16776206680,80622416976,389123999656,

%U 1885405316596,9167409871040,44717351734160,218762640192838,1073082055680180

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

%H Seiichi Manyama, <a href="/A262441/b262441.txt">Table of n, a(n) for n = 0..1000</a>

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

%F 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

%F a(n) = (n + 1)*hypergeom([1 - n, -n, n + 2], [3/2, 2], 1/4). - _Peter Luschny_, Mar 07 2022

%t 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 *)

%o (Maxima)

%o a(n):=sum(binomial(n,k)*binomial(n+k-2,n-k-1),k,0,n-1)/n;

%o A(x):=sum(a(n)*x^n,n,1,30);

%o taylor((1/x-1/A(x)),x,0,10);

%o (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

%o (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

%Y Cf. A109081, A161798.

%K nonn

%O 0,2

%A _Vladimir Kruchinin_, Sep 23 2015