A291697 a(n) = [x^n] Product_{k>=0} ((1 + x^(2*k+1))/(1 - x^(2*k+1)))^n.

1, 2, 8, 44, 256, 1512, 9056, 54896, 335872, 2069774, 12827888, 79875996, 499305472, 3131436856, 19694403520, 124165133424, 784478240768, 4965659813668, 31484486937512, 199923173603596, 1271192603065856, 8092551782518688, 51574780342740256, 329022223268286288, 2100934234342260736

Offset: 0

Author

Ilya Gutkovskiy, Aug 30 2017

Keywords

• nonn

Comments

From Peter Bala, Apr 18 2023: (Start)

The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.

Conjecture: the supercongruence a(p) == 2*p + 2 (mod p^3) holds for all primes p >= 5. Cf. A270919. (End)

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Main diagonal of A289522.

Cf. A080054, A008705, A270919, A361008, A362408.

Programs

Mathematica

Table[SeriesCoefficient[Product[((1 + x^(2 k + 1))/(1 - x^(2 k + 1)))^n, {k, 0, n}], {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[(QPochhammer[-x, x^2]/QPochhammer[x, x^2])^n, {x, 0, n}], {n, 0, 24}]
(* Calculation of constant d: *) 1/r /. FindRoot[{s == QPochhammer[-r*s, r^2*s^2] / QPochhammer[r*s, r^2*s^2], QPochhammer[r*s, r^2*s^2] + QPochhammer[r*s, r^2*s^2]*((QPolyGamma[0, Log[-r*s]/Log[r^2*s^2], r^2*s^2] - QPolyGamma[0, Log[r*s]/Log[r^2*s^2], r^2*s^2]) / Log[r^2*s^2]) + 2*r^2*s^2*Derivative[0, 1][QPochhammer][r*s, r^2*s^2] == 2*r^2*s*Derivative[0, 1][QPochhammer][-r*s, r^2*s^2]}, {r, 1/8}, {s, 1}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 04 2023 *)

Formula

a(n) = A289522(n,n).

a(n) ~ c * d^n / sqrt(n), where d = 6.52085730573545526010335599231748172235904... and c = 0.296494808714349908707366708893... - Vaclav Kotesovec, Aug 30 2017