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

%I A291697 #17 Oct 04 2023 03:41:18
%S A291697 1,2,8,44,256,1512,9056,54896,335872,2069774,12827888,79875996,
%T A291697 499305472,3131436856,19694403520,124165133424,784478240768,
%U A291697 4965659813668,31484486937512,199923173603596,1271192603065856,8092551782518688,51574780342740256,329022223268286288,2100934234342260736
%N A291697 a(n) = [x^n] Product_{k>=0} ((1 + x^(2*k+1))/(1 - x^(2*k+1)))^n.
%C A291697 From _Peter Bala_, Apr 18 2023: (Start)
%C A291697 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.
%C A291697 Conjecture: the supercongruence a(p) == 2*p + 2 (mod p^3) holds for all primes p >= 5. Cf. A270919. (End)
%H A291697 Vaclav Kotesovec, <a href="/A291697/b291697.txt">Table of n, a(n) for n = 0..1000</a>
%F A291697 a(n) = A289522(n,n).
%F A291697 a(n) ~ c * d^n / sqrt(n), where d = 6.52085730573545526010335599231748172235904... and c = 0.296494808714349908707366708893... - _Vaclav Kotesovec_, Aug 30 2017
%t A291697 Table[SeriesCoefficient[Product[((1 + x^(2 k + 1))/(1 - x^(2 k + 1)))^n, {k, 0, n}], {x, 0, n}], {n, 0, 24}]
%t A291697 Table[SeriesCoefficient[(QPochhammer[-x, x^2]/QPochhammer[x, x^2])^n, {x, 0, n}], {n, 0, 24}]
%t A291697 (* 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 *)
%Y A291697 Main diagonal of A289522.
%Y A291697 Cf. A080054, A008705, A270919, A361008, A362408.
%K A291697 nonn
%O A291697 0,2
%A A291697 _Ilya Gutkovskiy_, Aug 30 2017