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A333374 G.f.: Sum_{k>=1} (x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)/(1 - x^j)).

%I A333374 #21 Jan 09 2025 19:08:54
%S A333374 1,0,1,2,2,2,3,4,6,8,10,12,15,18,22,28,34,42,52,62,75,90,106,126,150,
%T A333374 176,208,246,288,338,397,462,538,626,724,838,968,1114,1282,1474,1690,
%U A333374 1936,2217,2532,2890,3296,3750,4264,4844,5492,6222,7042,7958,8986,10138
%N A333374 G.f.: Sum_{k>=1} (x^(k*(k+1)) * Product_{j=1..k} (1 + x^j)/(1 - x^j)).
%C A333374 The g.f. Sum_{k >= 1} x^(k*(k+1)) * Product_{j = 1..k} (1 + x^j)/(1 - x^j) = Sum_{k >= 1} x^(k*(k+1)) * Product_{j = 1..k} (1 + x^j)/(1 - x^j + 2*x^j) == Sum_{k >= 1} x^(k*(k+1)) (mod 2). It follows that a(n) is odd iff n = k*(k + 1) for some nonnegative integer k. - _Peter Bala_, Jan 04 2025
%H A333374 Vaclav Kotesovec, <a href="/A333374/b333374.txt">Table of n, a(n) for n = 0..10000</a>
%F A333374 Limit_{n->infinity} A066447(n) / a(n) = A058265 = (1 + (19+3*sqrt(33))^(1/3) + (19-3*sqrt(33))^(1/3))/3 = 1.839286755214... (the tribonacci constant).
%F A333374 Compare with: A306734(n) / A333179(n) -> A060006 (the plastic constant) and A003114(n) / A003106(n) -> A001622 (golden ratio).
%F A333374 a(n) ~ c * d^sqrt(n) / n^(3/4), where d = A376841 = 7.1578741786143524880205... = exp(2*sqrt(log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))) and c = 0.10511708841962944170826735560432... = (log(r)^2 - polylog(2, -r^2) + polylog(2, r^2))^(1/4) * sqrt(1/24 - sinh(arcsinh(sqrt(11)/4)/3) / (12*sqrt(11))) / sqrt(Pi), where r = A192918 = 0.54368901269207636157... is the real root of the equation r^2*(1+r) = 1-r. - _Vaclav Kotesovec_, Mar 17 2020, updated Oct 10 2024
%t A333374 nmax = 100; CoefficientList[Series[Sum[x^(n*(n+1))*Product[(1+x^k)/(1-x^k), {k, 1, n}], {n, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
%t A333374 nmax = 100; p = 1; s = 1; Do[p = Normal[Series[p*(1 + x^k)/(1 - x^k)*x^(2*k), {x, 0, nmax}]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]
%Y A333374 Cf. A001935, A001936, A003106, A053261, A066447, A003114, A306734, A333179.
%Y A333374 Cf. A192918, A376841.
%K A333374 nonn
%O A333374 0,4
%A A333374 _Vaclav Kotesovec_, Mar 17 2020