A262410 a(n) = (n+1)*Sum_{k=1..n} binomial(n-1,k-1)*binomial(n+2*k+2,k+1)/(n+k+2).

5, 37, 275, 2071, 15781, 121395, 940915, 7337560, 57507892, 452598884, 3574599205, 28316957579, 224901946395, 1790287826789, 14279629073403, 114097695427295, 913103420246956, 7317725618907700, 58719917176448820, 471733089071984376

Offset: 1

Author

Vladimir Kruchinin, Sep 22 2015

Keywords

• nonn

Crossrefs

Cf. A108447.

Programs

Mathematica

Table[(n + 1) Sum[Binomial[n - 1, k - 1] Binomial[n + 2 k + 2, k + 1]/(n + k + 2), {k, n}], {n, 20}] (* Michael De Vlieger, Sep 22 2015 *)

Maxima

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

PARI

a(n) = (n+1)*sum(k=1,n,((binomial(n-1,k-1) *binomial(n+2*k+2,k+1))/(n+k+2))) \\ Anders Hellström, Sep 22 2015

Formula

G.f.: B'(x)/B(x)-x*B'(x)-B(x), where B(x) is g.f. of A108447.

Recurrence: 2*n*(n+1)*(2*n + 1)*(74*n^3 - 183*n^2 + 129*n - 18)*a(n) = 2*n*(1184*n^5 - 1744*n^4 - 242*n^3 + 857*n^2 - 201*n + 26)*a(n-1) + 2*(n-2)*(296*n^5 + 8*n^4 - 410*n^3 + 35*n^2 + 29*n - 18)*a(n-2) - 5*(n-3)*(n-2)*(n+2)*(74*n^3 + 39*n^2 - 15*n + 2)*a(n-3). - Vaclav Kotesovec, Sep 22 2015