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Expansion of Product_{m>=1} (1+x^m)^3.
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%I #38 Oct 24 2024 12:33:32

%S 1,3,6,13,24,42,73,120,192,302,465,702,1046,1536,2226,3195,4536,6378,

%T 8896,12306,16896,23045,31224,42048,56310,75000,99384,131072,172071,

%U 224910,292774,379608,490338,631104,809472,1034814,1318707,1675344,2122176,2680602,3376728,4242432,5316562,6646272

%N Expansion of Product_{m>=1} (1+x^m)^3.

%H Seiichi Manyama, <a href="/A022568/b022568.txt">Table of n, a(n) for n = 0..1000</a>

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.

%H Mohammed L. Nadji and Moussa Ahmia, <a href="https://math.colgate.edu/~integers/y86/y86.pdf">Congruences for L-regular tripartitions for L in {2, 3}</a>, Integers (2024) Vol. 24, Art. No. A86. See p. 2.

%F a(n) ~ exp(Pi * sqrt(n)) / (8 * n^(3/4)) * (1 + (Pi/16 - 3/(8*Pi)) / sqrt(n)). - _Vaclav Kotesovec_, Mar 05 2015, extended Jan 16 2017

%F a(0) = 1, a(n) = (3/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 03 2017

%F G.f.: exp(3*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 06 2018

%t nmax=50; CoefficientList[Series[Product[(1+q^m)^3,{m,1,nmax}],{q,0,nmax}],q] (* _Vaclav Kotesovec_, Mar 05 2015 *)

%t nmax = 50; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 3; poly[[3]] = 3; poly[[4]] = 1; Do[Do[Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}];, {p, 1, 3}], {k, 2, nmax}]; poly (* _Vaclav Kotesovec_, Mar 31 2018 *)

%o (PARI) x='x+O('x^51); Vec(prod(m=1, 50, (1 + x^m)^3)) \\ _Indranil Ghosh_, Apr 03 2017

%o (Magma) Coefficients(&*[(1+x^m)^3:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // _G. C. Greubel_, Feb 26 2018

%Y Cf. A000009.

%Y Column k=3 of A286335.

%K nonn

%O 0,2

%A _N. J. A. Sloane_