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
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Aug 13 2015
STATUS
approved