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G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j)).
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%I #8 Oct 01 2024 05:23:14

%S 1,1,0,1,1,0,1,0,1,1,1,1,0,1,0,2,1,1,1,1,1,1,2,0,2,1,2,1,2,1,2,2,1,2,

%T 1,3,2,3,1,3,1,3,2,3,2,3,3,2,4,3,4,2,4,2,5,3,5,2,5,3,5,4,4,5,5,5,5,6,

%U 4,7,4,7,4,8,4,8,5,8,6,8,6,9,7,8,8,8,9,8,10

%N G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j)).

%H Vaclav Kotesovec, <a href="/A376632/b376632.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ sqrt(1 + 3/sqrt(5)) * exp(Pi*sqrt(n/30)) / (4*sqrt(n)).

%t nmax = 100; CoefficientList[Series[Sum[x^(k^2)*Product[1+x^(2*j), {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

%t nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k))*x^(2*k - 1)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

%Y Cf. A306734, A376627, A376631.

%K nonn

%O 0,16

%A _Vaclav Kotesovec_, Sep 30 2024