A185655 a(n) = Sum_{k=0..n} binomial(n+k, k)*binomial(n+k+1, k+1)/(n+1).

1, 4, 27, 236, 2375, 26090, 304241, 3704860, 46622655, 602035556, 7937288062, 106451074614, 1448267147717, 19944962832826, 277565209168861, 3898075200816892, 55182857681572655, 786731161113510584

Offset

0,2

Comments

The function B(x,r) = x*Sum_{n>=0} b(n,r)*x^n, where

b(n,r) = Sum_{k=0..n} binomial(n+k, k)*binomial(n+k+1, r*k+1)/(n+1), satisfies

B(x/(1+x) - x^r, r) = x for all positive integer r except at r=1;

B(x,1)/x is the generating function of this sequence.

Links

G. C. Greubel, Table of n, a(n) for n = 0..830

Formula

Recurrence: 2*(n+1)^2*(2*n + 1)*(3*n - 2)*(7*n - 2)*a(n) = (1365*n^5 - 607*n^4 - 821*n^3 + 411*n^2 + 80*n - 44)*a(n-1) - 4*(n-2)*(2*n - 1)^2*(3*n + 1)*(7*n + 5)*a(n-2). - Vaclav Kotesovec, Nov 27 2017

a(n) ~ 2^(4*n+3) / (3*Pi*n^2). - Vaclav Kotesovec, Nov 27 2017

Example

G.f.: A(x) = 1 + 4*x + 27*x^2 + 236*x^3 + 2375*x^4 + 26090*x^5 +...
Let G(x*A(x)) = x, then the series reversion of x*A(x) begins:
G(x) = x - 4*x^2 + 5*x^3 - 16*x^4 - 12*x^5 - 218*x^6 - 1197*x^7 - 8974*x^8 - 65582*x^9 - 503614*x^10 - 3956461*x^11 - ...
Does G(x) satisfy a nice functional equation?

Mathematica

Table[Sum[Binomial[n + k, k]*Binomial[n + k + 1, k + 1]/(n + 1), {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Jul 09 2017 *)

Prog

(PARI) {a(n)=sum(k=0, n, binomial(n+k, k)*binomial(n+k+1, k+1))/(n+1)}

Crossrefs

Sequence in context: A160883 A362274 A328978 * A181146 A303559 A161120

Adjacent sequences: A185652 A185653 A185654 * A185656 A185657 A185658

Keyword

nonn

Author

Paul D. Hanna, Feb 15 2011

Status

approved