OFFSET
0,2
COMMENTS
A fast algorithm follows from Theorem 5 of Brent et al. article.
LINKS
Colin Barker, Table of n, a(n) for n = 0..800
Richard P. Brent, Hideyuki Ohtsuka, Judy-anne H. Osborn, and Helmut Prodinger, Some Binomial Sums Involving Absolute Values, Journal of Integer Sequences, Vol. 19 (2016), Article 16.3.7; arXiv preprint, arXiv:1411.1477v2 [math.CO], 2014-2016.
Index entries for linear recurrences with constant coefficients, signature (48,-768,4096).
FORMULA
a(n) = Sum_{k=-n..n} (Sum_{l=-n..n} binomial(2*n, n+k)*binomial(2*n, n+l)*abs(k^2 - l^2)^2).
From Colin Barker, Feb 11 2016: (Start)
a(n) = 2^(4*n-1)*n*(2*n-1).
a(n) = 48*a(n-1)-768*a(n-2)+4096*a(n-3) for n>2.
G.f.: 8*x*(1+48*x) / (1-16*x)^3. (End)
From Amiram Eldar, Oct 23 2025: (Start)
Sum_{n>=1} 1/a(n) = arctanh(1/4) - 2*log(16/15).
Sum_{n>=1} (-1)^(n+1)/a(n) = arctan(1/4) - 2*log(17/16). (End)
MATHEMATICA
a[n_] := 2^(4*n-1)*n*(2*n-1); Array[a, 20, 0] (* Amiram Eldar, Oct 23 2025 *)
PROG
(PARI) a(n) = sum(k=-n, n, sum(l=-n, n, binomial(2*n, n+k)*binomial(2*n, n+l)*abs(k^2 - l^2)^2));
(PARI) concat(0, Vec(8*x*(1+48*x)/(1-16*x)^3 + O(x^20))) \\ Colin Barker, Feb 11 2016
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Richard P. Brent, Jan 27 2016
STATUS
approved