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%I #29 Jun 30 2023 02:42:13
%S 1,7,131,3067,79459,2181257,62165039,1818812387,54257991011,
%T 1642977121597,50344383988381,1557608560147757,48577698917598031,
%U 1525245771206644117,48165918788138198759,1528611371067309862067
%N Quotient of de Bruijn sums S(4,n)/S(2,n).
%C The de Bruijn sum S(s,n) = Sum_{k=0..2n} (-1)^(k+n) binomial(2n,k)^s.
%C Also the n-th term of the crystal ball sequence for the A_{2n} lattice.
%D 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).
%H Vincenzo Librandi, <a href="/A099601/b099601.txt">Table of n, a(n) for n = 0..200</a>
%F 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
%F a(n) ~ (1+sqrt(2))^(4*n+3/2) / (2^(9/4)*Pi*n). - _Vaclav Kotesovec_, Mar 07 2014
%F 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
%F a(n) = hypergeom([-2*n, -n, 2*n+1], [1, 1], 1). - _Peter Luschny_, Feb 13 2018
%F From _Peter Bala_, Dec 21 2020: (Start)
%F a(n) = Sum_{k = 0..n} C(2*n,n-k)^2 * C(2*n+k,k). Cf. A005258.
%F a(n) = Sum_{k = 0..n} (-1)^(n+k)*C(2*n,n-k)*C(2*n+k,k)^2.
%F a(n) = C(2*n,n)^2 * hypergeom([-n, -n, 2*n+1], [n+1, n+1], 1).
%F a(n) = (-1)^n* C(2*n,n) * hypergeom([-n, 2*n+1, 2*n+1], [1, n+1], 1). (End)
%F 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
%F a(n) = A108625(2*n, n). - _Peter Bala_, Jun 20 2023
%e A003215(1)=7=a(1), A008384(2)=131=a(2), A008388(3)=3067=a(3), ...
%p a := n -> hypergeom([-2*n, -n, 2*n+1], [1, 1], 1):
%p seq(simplify(a(n)), n=0..15); # _Peter Luschny_, Feb 13 2018
%t 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 *)
%o (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))
%Y Cf. A000984, A003215, A005258, A008384, A008388, A008392, A008396, A050983, A108625.
%K nonn,easy
%O 0,2
%A _Michael Somos_, Oct 24 2004
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