A329021 a(n) is the constant term in the expansion of ( Sum_{k=1..n} x_k^(2*k-1) + x_k^(-(2*k-1)) )^(2*n).

1, 2, 44, 4332, 1012664, 432457640, 293661065788, 290711372717976, 395320344293410544, 707125993042984343136, 1609908874238209683872480, 4545914321591993313415189408, 15591582457233317184439165505544, 63847180690107503874880321918389332

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..100

Formula

a(n) = A077045(2*n).

a(n) = Sum_{k=0..floor((2*n-1)/2)} (-1)^k*binomial(2*n,k)*binomial((2*n+1)*n-2*n*k-1,(2*n-1)*n-2*n*k) for n > 0.

a(n) ~ sqrt(3) * 2^(2*n - 1) * n^(2*n - 3/2) / sqrt(Pi). - Vaclav Kotesovec, Feb 04 2022

Example

(x + 1/x)^2 = x^2 + 2 + 1/x^2. So a(1) = 2.

Mathematica

a[0] = 1; a[n_] := Sum[(-1)^k * Binomial[2*n, k] * Binomial[(2*n + 1)*n - 2*n*k - 1, (2*n - 1)*n - 2*n*k], {k, 0, Floor[n - 1/2]}]; Array[a, 14, 0] (* Amiram Eldar, May 06 2021 *)

Prog

(PARI) {a(n) = polcoef((sum(k=1, n, x^(2*k-1)+x^(-(2*k-1))))^(2*n), 0)}
(PARI) {a(n) = if(n==0, 1, sum(k=0, (2*n-1)\2, (-1)^k*binomial(2*n, k)*binomial((2*n+1)*n-2*n*k-1, (2*n-1)*n-2*n*k)))}

Crossrefs

Main diagonal of A329020.

Cf. A077045.

Sequence in context: A161722 A290879 A054914 * A275307 A356484 A291808

Adjacent sequences: A329018 A329019 A329020 * A329022 A329023 A329024

Keyword

nonn

Author

Seiichi Manyama, Nov 02 2019

Status

approved