A262607 Sum_{k=0..n} ((k+1)*binomial(n+1,k)*binomial(2*n-k,n))/(n+1).

1, 3, 11, 47, 219, 1075, 5459, 28383, 150131, 804515, 4355163, 23768079, 130572363, 721247571, 4002344355, 22296869823, 124633584099, 698707769923, 3927060020651, 22121780745711, 124865811262139, 706065855417203, 3998950848888051

Offset

0,2

Links

Table of n, a(n) for n=0..22.

D. Drake, Bijections from Weighted Dyck Paths to Schröder Paths, J. Int. Seq. 13 (2010) # 10.9.2, Table 1.

Formula

G.f.: (-2*x^2+7*x-1)/(2*x*sqrt(x^2-6*x+1))+1/(2*x)-1.

G.f. satisfies -A'(x)/A(x)+A'(x)/x, where A(x)/x is g.f. of A155069

-(n+1)*(2*n^2+5*n-6)*a(n) +6*(2*n^3+6*n^2-11*n+4)*a(n-1) -(n-2)*(2*n^2+9*n+1)*a(n-2)=0. - R. J. Mathar, Jul 21 2017

a(n) ~ (1 + sqrt(2))^(2*n) / (2^(5/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Jul 10 2021

Mathematica

Table[Sum[(k + 1) Binomial[n + 1, k] Binomial[2 n - k, n]/(n + 1), {k,
0, n}], {n, 0, 22}] (* Michael De Vlieger, Sep 26 2015 *)

Prog

(Maxima)
A(x):=x*(3-x-sqrt(1-6*x+x^2))/2;
taylor(-diff(A(x), x)/A(x)+diff(A(x), x, 1)/x, x, 0, 27);

Crossrefs

Cf. A155069.

Sequence in context: A301770 A308227 A247313 * A059284 A118927 A217216

Adjacent sequences: A262604 A262605 A262606 * A262608 A262609 A262610

Keyword

nonn

Author

Vladimir Kruchinin, Sep 26 2015

Status

approved