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Expansion of exp(Sum_{k>=1} ( Sum_{d|k, d odd} d^k ) * x^k/k).
1

%I #6 Sep 07 2018 04:47:14

%S 1,1,1,10,10,635,797,118446,124071,43174194,45404910,25982930761,

%T 26443958420,23324558686914,23640266984002,29216576615057082,

%U 29447535265299613,48690644491136860817,48980258924147884960,104176334607664412086539,104636388540330684649083,278323070872780066332365486

%N Expansion of exp(Sum_{k>=1} ( Sum_{d|k, d odd} d^k ) * x^k/k).

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

%t nmax = 21; CoefficientList[Series[Exp[Sum[Sum[Mod[d, 2] d^k, {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]

%t nmax = 21; CoefficientList[Series[Product[1/(1 - (2 k - 1)^(2 k - 1) x^(2 k - 1))^(1/(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]

%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Mod[d, 2] d^k, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 21}]

%Y Cf. A000009, A023881, A206303, A262811, A292919, A318969.

%K nonn

%O 0,4

%A _Ilya Gutkovskiy_, Sep 06 2018