A269877 - OEIS

%I #33 Jan 12 2025 12:56:07

%S 0,8,121728,77214720,12676235264,1090372239360,64922717257728,

%T 3052335748087808,121762580539637760,4304417014325182464,

%U 138706918527488491520,4154140250223566389248,117243264067548833906688,3150495258536853477785600,81236017376284183797694464

%N 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.

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

%H Vincenzo Librandi, <a href="/A269877/b269877.txt">Table of n, a(n) for n = 0..800</a>

%H Richard P. Brent, Hideyuki Ohtsuka, Judy-anne H. Osborn, Helmut Prodinger, <a href="http://arxiv.org/abs/1411.1477">Some binomial sums involving absolute values</a>, arXiv:1411.1477 [math.CO], 2016, page 11.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (112,-5376,143360,-2293760,22020096,-117440512,268435456).

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

%F 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).

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

%t Table[2^(4 n - 3) n (2 n - 1) (900 n^4 - 4500 n^3 + 8895 n^2 - 8055 n + 2764), {n, 0, 15}]

%t LinearRecurrence[{112,-5376,143360,-2293760,22020096,-117440512,268435456},{0,8,121728,77214720,12676235264,1090372239360,64922717257728},20] (* _Harvey P. Dale_, Oct 28 2023 *)

%o (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]];

%Y Cf. A001025, A268148, A268152.

%K nonn,easy

%O 0,2

%A _Vincenzo Librandi_, May 10 2016