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A022598 Expansion of Product_{m>=1} (1+q^m)^(-3).

%I A022598 #36 Mar 27 2025 15:06:42
%S A022598 1,-3,3,-4,9,-12,15,-21,30,-43,54,-69,94,-123,153,-193,252,-318,391,
%T A022598 -486,609,-754,918,-1119,1376,-1680,2019,-2432,2946,-3540,4220,-5034,
%U A022598 6015,-7157,8463,-9999,11835,-13956,16374,-19206,22542,-26376,30750,-35829,41745
%N A022598 Expansion of Product_{m>=1} (1+q^m)^(-3).
%C A022598 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A022598 T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, q_2^3.
%H A022598 Seiichi Manyama, <a href="/A022598/b022598.txt">Table of n, a(n) for n = 0..1000</a>
%H A022598 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. 13.
%H A022598 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A022598 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A022598 Expansion of chi(-x)^3 = phi(-x) / psi(x) in powers of x where phi(), psi(), chi() are Ramanujan theta functions. - _Michael Somos_, Aug 09 2015
%F A022598 Expansion of q^(1/8) * (eta(q) / eta(q^2))^3 in powers of q. - _Michael Somos_, Apr 24 2015
%F A022598 Euler transform of period 2 sequence [ -3, 0, ...]. - _Michael Somos_, Aug 09 2015
%F A022598 Convolution cube of A081362. - _Michael Somos_, Apr 24 2015
%F A022598 Convolution inverse of A022568. - _Michael Somos_, Aug 09 2015
%F A022598 a(n) ~ (-1)^n * exp(Pi*sqrt(n/2)) / (2^(7/4) * n^(3/4)). - _Vaclav Kotesovec_, Aug 27 2015
%F A022598 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 A022598 G.f.: exp(-3*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 06 2018
%e A022598 G.f. = 1 - 3*x + 3*x^2 - 4*x^3 + 9*x^4 - 12*x^5 + 15*x^6 - 21*x^7 + 30*x^8 + ...
%e A022598 G.f. = 1/q - 3*q^7 + 3*q^15 - 4*q^23 + 9*q^31 - 12*q^39 + 15*q^47 - 21*q^55 + ...
%t A022598 a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[x^2])^3, {x, 0, n}]; (* _Michael Somos_, Feb 22 2015 *)
%t A022598 nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 27 2015 *)
%o A022598 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^3, n))};
%Y A022598 Cf. A022568, A081362.
%Y A022598 Column k=3 of A286352.
%K A022598 sign
%O A022598 0,2
%A A022598 _N. J. A. Sloane_