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Expansion of Product_{k>=1} 1 / (1 - Sum_{j>=1} j * x^(k*j)).
2

%I #11 Nov 07 2019 04:08:44

%S 1,1,4,10,29,72,200,510,1364,3546,9348,24400,64090,167562,439200,

%T 1149360,3010349,7879832,20633304,54014950,141422328,370239300,

%U 969323000,2537696160,6643839400,17393731933,45537549048,119218684970,312119004990,817137724392,2139295489200,5600747143950

%N Expansion of Product_{k>=1} 1 / (1 - Sum_{j>=1} j * x^(k*j)).

%C Euler transform of A032198.

%F G.f.: Product_{k>=1} 1 / (1 - x^k / (1 - x^k)^2).

%F G.f.: exp(Sum_{k>=1} ( Sum_{d|k} 1 / (d * (1 - x^(k/d))^(2*d)) ) * x^k).

%F G.f.: Product_{k>=1} 1 / (1 - x^k)^A032198(k).

%F G.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = g.f. of A088305.

%F a(n) ~ phi^(2*n-1), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Nov 07 2019

%t nmax = 31; CoefficientList[Series[Product[1/(1 - Sum[j x^(k j), {j, 1, nmax}]), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 31; CoefficientList[Series[Product[1/(1 - x^k/(1 - x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A006951, A032198, A088305, A162891, A258210, A329157, A329163.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Nov 06 2019