OFFSET
0,2
COMMENTS
The de Bruijn sum S(s,n) = Sum_{k=0..2n} (-1)^(k+n) binomial(2n,k)^s.
Also the n-th term of the crystal ball sequence for the A_{2n} lattice.
REFERENCES
G. E. Andrews, Application of SCRATCHPAD to Problems in Special Functions and Combinatorics, in Trends in Computer Algebra, Springer-Verlag, 1988, pp. 158-166 MR0935413 (89c:05010).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
Recurrence: n^2*(2*n-1)^2*(48*n^2 - 126*n + 83)*a(n) = (6528*n^6 - 30192*n^5 + 55040*n^4 - 50406*n^3 + 24465*n^2 - 5985*n + 585)*a(n-1) - (n-1)^2*(2*n-3)^2*(48*n^2 - 30*n + 5)*a(n-2). - Vaclav Kotesovec, Mar 07 2014
a(n) ~ (1+sqrt(2))^(4*n+3/2) / (2^(9/4)*Pi*n). - Vaclav Kotesovec, Mar 07 2014
a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*n+k,2*n-k)*binomial(2*k,k). - Ilya Gutkovskiy, Nov 24 2017
a(n) = hypergeom([-2*n, -n, 2*n+1], [1, 1], 1). - Peter Luschny, Feb 13 2018
From Peter Bala, Dec 21 2020: (Start)
a(n) = Sum_{k = 0..n} C(2*n,n-k)^2 * C(2*n+k,k). Cf. A005258.
a(n) = Sum_{k = 0..n} (-1)^(n+k)*C(2*n,n-k)*C(2*n+k,k)^2.
a(n) = C(2*n,n)^2 * hypergeom([-n, -n, 2*n+1], [n+1, n+1], 1).
a(n) = (-1)^n* C(2*n,n) * hypergeom([-n, 2*n+1, 2*n+1], [1, n+1], 1). (End)
a(n) = [x^n] 1/(1 - x)*( Legendre_P(m*n,(1 + x)/(1 - x)) ) at m = 2. At m = 1 we get the Apéry numbers A005258. - Peter Bala, Dec 23 2020
a(n) = A108625(2*n, n). - Peter Bala, Jun 20 2023
MAPLE
a := n -> hypergeom([-2*n, -n, 2*n+1], [1, 1], 1):
seq(simplify(a(n)), n=0..15); # Peter Luschny, Feb 13 2018
MATHEMATICA
Table[Sum[(-1)^k*Binomial[2*n, k]^4, {k, 0, 2*n}] / Sum[(-1)^k*Binomial[2*n, k]^2, {k, 0, 2*n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 07 2014 *)
PROG
(PARI) a(n)=if(n<0, 0, sum(k=0, 2*n, (-1)^k*binomial(2*n, k)^4)/ sum(k=0, 2*n, (-1)^k*binomial(2*n, k)^2))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Oct 24 2004
STATUS
approved