A261266 Expansion of ((x-1/2)*(1/sqrt(8*x^2-8*x+1)+1)-x)/(x-1).

1, 2, 8, 44, 264, 1632, 10256, 65200, 418144, 2700224, 17534208, 114380928, 748988928, 4920379648, 32413343488, 214038123264, 1416349369856, 9389756730368, 62352450867200, 414660440811520, 2761261291024384

Offset

0,2

Links

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

D. Kruchinin and V. Kruchinin, A Generating Function for the Diagonal T2n,n in Triangles, Journal of Integer Sequence, Vol. 18 (2015), article 15.4.6.

Formula

a(n) = Sum_{l=0..n}(C(n,l)*Sum_{i=l..n}(C(i+l-1,i)*C(n-l,n-i))).

a(n) ~ 2^((3*n-1)/2) * (1+sqrt(2))^(n-1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Aug 13 2015

D-finite with recurrence: n*a(n) +(-11*n+8)*a(n-1) +2*(17*n-28)*a(n-2) +8*(-5*n+12)*a(n-3) +16*(n-3)*a(n-4)=0. - R. J. Mathar, Jan 25 2020

Maple

S := (n, k) -> simplify(binomial(2*k-1, k)*hypergeom([2*k, k-n], [k+1], -1)):
a := (n) -> add(binomial(n, k)*S(n, k), k=0..n):
seq(a(n), n=0..20); # Peter Luschny, Aug 13 2015

Mathematica

CoefficientList[Series[((x - 1/2)*(1/Sqrt[8*x^2 - 8*x + 1] + 1) - x)/(x - 1), {x, 0, 50}], x] (* G. C. Greubel, Jun 04 2017 *)

Prog

(Maxima)
a(n):=sum(binomial(n, l)*sum(binomial(i+l-1, i)*binomial(n-l, n-i), i, l, n), l, 0, n);
(PARI) x='x+O('x^50); Vec(((x-1/2)*(1/sqrt(8*x^2-8*x+1)+1)-x)/(x-1)) \\ G. C. Greubel, Jun 04 2017

Crossrefs

Cf. A118376.

Sequence in context: A047851 A177260 A121747 * A014508 A141147 A379328

Adjacent sequences: A261263 A261264 A261265 * A261267 A261268 A261269

Keyword

nonn

Author

Vladimir Kruchinin, Aug 13 2015

Status

approved