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A110276 Convolution of large Schroeder numbers and central binomial coefficients. 2
1, 4, 16, 66, 280, 1218, 5422, 24666, 114540, 542278, 2614178, 12814102, 63772982, 321754290, 1643263134, 8483485886, 44214343344, 232362906298, 1230090777342, 6553657204178, 35113127086114, 189062666857686, 1022459506515674 (list; graph; refs; listen; history; text; internal format)
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

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Last modified February 3 17:00 EST 2023. Contains 360044 sequences. (Running on oeis4.)