A278646 a(n) = ((2*n+1)/(n+1))*Sum_{j=0..n/2} binomial(n+1, j)*binomial(n-j-1, n-2*j).

1, 0, 5, 7, 27, 66, 195, 540, 1547, 4408, 12663, 36455, 105325, 305046, 885573, 2576007, 7506675, 21909720, 64039119, 187418205, 549141249, 1610708190, 4729015305, 13896677718, 40870234629, 120290710176, 354292339625, 1044178154305

Offset

0,3

Links

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

Formula

G.f.: -(2*x*sqrt(-3*x^2-2*x+1)-2*x-2)/(-3*x^3+sqrt(-3*x^2-2*x+1)*(x^2+2*x+1)-5*x^2-x+1).

a(n) = (2*n + 1)*3F2(1-n/2,3/2-n/2,-n; 2,2-n; 4). - Ilya Gutkovskiy, Nov 25 2016

a(n) ~ 3^(n+3/2)/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Nov 27 2016

Mathematica

Table[(2 n + 1)/(n + 1) Sum[Binomial[n + 1, j] Binomial[n - j - 1, n - 2 j], {j, 0, Floor[n/2]}], {n, 0, 27}] (* or *)
CoefficientList[Series[-(2 x # - 2 x - 2)/(-3 x^3 + # (x^2 + 2 x + 1) - 5 x^2 - x + 1) &@ Sqrt[-3 x^2 - 2 x + 1], {x, 0, 27}], x] (* Michael De Vlieger, Nov 24 2016 *)

Prog

(Maxima) a(n):=((2*n+1)*sum(binomial(n+1, j)*binomial(n-j-1, n-2*j), j, 0, n/2))/(n+1);
(PARI) a(n) = (2*n+1)*sum(j=0, n\2, binomial(n+1, j)*binomial(n-j-1, n-2*j))/(n+1); \\ Michel Marcus, Dec 03 2016

Crossrefs

Cf. A005043.

Sequence in context: A166100 A135606 A051845 * A029668 A144392 A268701

Adjacent sequences: A278643 A278644 A278645 * A278647 A278648 A278649

Keyword

nonn

Author

Vladimir Kruchinin, Nov 24 2016

Status

approved