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A366104 G.f. ( Chi(sqrt(x))^4 + Chi(-sqrt(x))^4 )/2, where Chi(x) = Product_{k >= 0} 1 + x^(2*k+1) is the g.f. of A000700.

%I A366104 #17 Jun 29 2025 07:36:55
%S A366104 1,6,17,38,84,172,325,594,1049,1796,3005,4912,7877,12430,19309,29580,
%T A366104 44766,66978,99150,145374,211242,304382,435194,617674,870651,1219352,
%U A366104 1697283,2348888,3232919,4426546,6030872,8177986,11039633,14838518,19862613,26482878,35175989,46552818,61393694
%N A366104 G.f. ( Chi(sqrt(x))^4 + Chi(-sqrt(x))^4 )/2, where Chi(x) = Product_{k >= 0} 1 + x^(2*k+1) is the g.f. of A000700.
%C A366104 Compare with A224916 with g.f. ( Chi(sqrt(x))^4 - Chi(-sqrt(x))^4 )/(8*sqrt(x)),
%C A366104 A069910 with g.f. ( Chi(sqrt(x)) + Chi(-sqrt(x)) )/2,
%C A366104 A069911 with g.f. ( Chi(sqrt(x)) - Chi(-sqrt(x)) )/2,
%C A366104 A226622 with g.f. ( Chi(sqrt(x))^2 + Chi(-sqrt(x))^2 )/2 and
%C A366104 A226635 with g.f. ( Chi(sqrt(x))^2 - Chi(-sqrt(x))^2 )/(4*sqrt(x)),
%C A366104 Jacobi's "aequatio identica satis abstrusa" is the identity ( Chi(sqrt(x))^8 - Chi(-sqrt(x))^8 )/(16*sqrt(x)) =  Product_{k >= 1} (1 + x^k)^8.
%F A366104 G.f.: Product_{k >= 1} (1 + x^(2*k))^2*(1 + x^(2*k-1))^6.
%F A366104 G.f.: x^(1/12) * eta(x^2)^10 * eta(x^4)^2 / ( eta(x) * eta(x^4) )^6.
%F A366104 a(n) ~ exp(2*Pi*sqrt(n/3)) / (4 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 29 2025
%p A366104 with(QDifferenceEquations):
%p A366104  seq(coeff((1/2)*expand(QPochhammer(-q,q^2,40)^4 + QPochhammer(q,q^2,40)^4), q, 2*n), n = 0..40);
%p A366104 #alternative program
%p A366104 seq(coeff(expand(QPochhammer(-q^2, q^2, 20)^2 * QPochhammer(-q, q^2, 20)^6), q, n), n = 0..40);
%t A366104 nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k))^2 * (1 + x^(2*k-1))^6, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 29 2025 *)
%Y A366104 Cf. A000700, A014705, A069910, A069911, A224916, A226622, A226635.
%K A366104 nonn,easy
%O A366104 0,2
%A A366104 _Peter Bala_, Sep 29 2023