A110276 - OEIS

%I #21 Sep 08 2022 08:45:19

%S 1,4,16,66,280,1218,5422,24666,114540,542278,2614178,12814102,

%T 63772982,321754290,1643263134,8483485886,44214343344,232362906298,

%U 1230090777342,6553657204178,35113127086114,189062666857686,1022459506515674

%N Convolution of large Schroeder numbers and central binomial coefficients.

%H G. C. Greubel, <a href="/A110276/b110276.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (1-x-sqrt(1-6*x+x^2))/(2*x*sqrt(1-4*x)). - corrected by _Georg Fischer_, Apr 09 2020

%F 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) ).

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

%F a(n) ~ sqrt(4 + sqrt(2)) * (1 + sqrt(2))^(2*n + 2) / (2*sqrt(7*Pi)*n^(3/2)). - _Vaclav Kotesovec_, Sep 14 2021

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

%o (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

%o (Sage)

%o def A110276_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( (1-x-sqrt(1-6*x+x^2))/(2*x*sqrt(1-4*x)) ).list()

%o A110276_list(30)

%o (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

%Y Cf. A000984, A006318.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jul 18 2005