A269877 a(n) = 2^(4*n-3)*n*(2*n-1)*(900*n^4-4500*n^3+8895*n^2-8055*n+2764), a closed form for a double binomial sum involving absolute values.

0, 8, 121728, 77214720, 12676235264, 1090372239360, 64922717257728, 3052335748087808, 121762580539637760, 4304417014325182464, 138706918527488491520, 4154140250223566389248, 117243264067548833906688, 3150495258536853477785600, 81236017376284183797694464

Offset: 0

Author

Vincenzo Librandi, May 10 2016

Keywords

• nonn
• easy

Comments

A fast algorithm follows from Theorem 5 of Brent et al. article.

Links

• Vincenzo Librandi, 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:1411.1477 [math.CO], 2016, page 11.
• Index entries for linear recurrences with constant coefficients, signature (112,-5376,143360,-2293760,22020096,-117440512,268435456).

Crossrefs

Cf. A001025, A268148, A268152.

Programs

Magma

[2^(4*n-3)*n*(2*n-1)*(900*n^4-4500*n^3+8895*n^2-8055*n+2764): n in [0..20]];

Mathematica

Table[2^(4 n - 3) n (2 n - 1) (900 n^4 - 4500 n^3 + 8895 n^2 - 8055 n + 2764), {n, 0, 15}]
LinearRecurrence[{112,-5376,143360,-2293760,22020096,-117440512,268435456},{0,8,121728,77214720,12676235264,1090372239360,64922717257728},20] (* Harvey P. Dale, Oct 28 2023 *)

Formula

G.f.: 8*x*(1 + 15104*x + 7953024*x^2 + 585181184*x^3 + 8538456064*x^4 + 19750453248*x^5)/(1-16*x)^7.

a(n) = Sum_{k=-n..n} (Sum_{l=-n..n} binomial(2*n, n+k)*binomial(2*n, n+l)*(k^2 - l^2)^6).

a(n) = 2^(4*n-3)*n*(2*n-1)*(900*n^4-4500*n^3+8895*n^2-8055*n+2764).