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

Author

Paul Barry, Jul 18 2005

Keywords

• easy
• nonn

Links

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

Crossrefs

Cf. A000984, A006318.

Programs

Magma

R:=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

Mathematica

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

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

Sage

def A110276_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x-sqrt(1-6*x+x^2))/(2*x*sqrt(1-4*x)) ).list()
A110276_list(30)

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