A268148 A double binomial sum involving absolute values.

0, 8, 768, 30720, 917504, 23592960, 553648128, 12213813248, 257698037760, 5257039970304, 104453604638720, 2031897488130048, 38843546786070528, 731834939447705600, 13618885273168379904, 250760427251989217280, 4574792530279968800768, 82788987402808467652608

Offset: 0

Author

Richard P. Brent, Jan 27 2016

Keywords

• easy
• nonn

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, Helmut Prodinger, Some binomial sums involving absolute values, arXiv:1411.1477v2 [math.CO], 2016.
• Index entries for linear recurrences with constant coefficients, signature (48,-768,4096).

Crossrefs

Cf. A000984, A002894, A166337, A254408.

Programs

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

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)