A262910 a(n) = Sum_{k=0..n} binomial(k+n-1,k)*binomial(k+n,2*k).

1, 2, 10, 59, 366, 2337, 15205, 100235, 667222, 4474733, 30188335, 204646532, 1392850785, 9511878729, 65144238981, 447263887479, 3077459618886, 21215286546705, 146500755609415, 1013180180867125, 7016536189029551, 48650933146617728, 337709155342663620

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1176

Formula

G.f.: A(x) = x*B'(x)/B(x), where B(x)/x is g.f. of A007863.

Recurrence: 5*(n-1)*n*(35*n^2 - 143*n + 138)*a(n) = 2*(n-1)*(630*n^3 - 2889*n^2 + 3746*n - 1200)*a(n-1) - 2*(70*n^4 - 426*n^3 + 811*n^2 - 589*n + 150)*a(n-2) + 2*(n-3)*(2*n - 3)*(35*n^2 - 73*n + 30)*a(n-3). - Vaclav Kotesovec, Oct 04 2015

a(n) = hypergeom([-n, n, n+1], [1/2, 1], -1/4). - Peter Luschny, Oct 08 2015

a(n) = A155112(2n,n). - Alois P. Heinz, Sep 29 2022

Maple

a := n -> hypergeom([-n, n, n+1], [1/2, 1], -1/4):
seq(round(evalf(a(n), 32)), n=0..21); # Peter Luschny, Oct 08 2015

Mathematica

Table[Sum[Binomial[k+n-1, k]*Binomial[k+n, 2*k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 04 2015 *)

Prog

(Maxima)
B(x):=sum(sum(binomial(i+n-1, i)*binomial(i+n, 2*i+1), i, 0, n-1)/n*x^n, n, 1, 30);
taylor(x*diff(B(x), x)/B(x), x, 0, 20);
(PARI) a(n) = sum(k=0, n, binomial(k+n-1, k)*binomial(k+n, 2*k));
vector(50, n, a(n-1)) \\ Altug Alkan, Oct 04 2015

Crossrefs

Cf. A007863, A155112.

Sequence in context: A309955 A340987 A186758 * A370281 A351502 A202482

Adjacent sequences: A262907 A262908 A262909 * A262911 A262912 A262913

Keyword

nonn

Author

Vladimir Kruchinin, Oct 04 2015

Status

approved