A110276 Convolution of large Schroeder numbers and central binomial coefficients.

1, 4, 16, 66, 280, 1218, 5422, 24666, 114540, 542278, 2614178, 12814102, 63772982, 321754290, 1643263134, 8483485886, 44214343344, 232362906298, 1230090777342, 6553657204178, 35113127086114, 189062666857686, 1022459506515674

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

G.f.: (1-x-sqrt(1-6*x+x^2))/(2*x*sqrt(1-4*x)). - corrected by Georg Fischer, Apr 09 2020

a(n) = Sum_{k=0..n} C(2*k, k)*( Sum_{j=0..n-k} C(n-k+j, n-k)*C(n-k, j)/(j+1) ).

a(n) = Sum_{k=0..n} A000984(k)*A006318(n-k).

a(n) ~ sqrt(4 + sqrt(2)) * (1 + sqrt(2))^(2*n + 2) / (2*sqrt(7*Pi)*n^(3/2)). - Vaclav Kotesovec, Sep 14 2021

Mathematica

CoefficientList[Series[(1-x-(Sqrt[1-6*x+x^2]))/(2x*Sqrt[1-4*x]), {x, 0, 30}] (* Georg Fischer, Apr 09 2020 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-x-Sqrt(1-6*x+x^2))/(2*x*Sqrt(1-4*x)) )); // G. C. Greubel, Sep 24 2021
(Sage)
def A110276_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1-x-sqrt(1-6*x+x^2))/(2*x*sqrt(1-4*x)) ).list()
A110276_list(30)
(PARI) a(n) = sum(k=0, n, binomial(2*k, k)*sum(j=0, n-k, binomial(n-k+j, n-k)*binomial(n-k, j)/(j+1))); \\ Michel Marcus, Sep 25 2021

Crossrefs

Cf. A000984, A006318.

Sequence in context: A082307 A099782 A109034 * A026883 A349730 A151242

Adjacent sequences: A110273 A110274 A110275 * A110277 A110278 A110279

Keyword

easy,nonn

Author

Paul Barry, Jul 18 2005

Status

approved