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A022569 Expansion of Product_{m>=1} (1+x^m)^4.

%I A022569 #43 Feb 16 2025 08:32:34
%S A022569 1,4,10,24,51,100,190,344,601,1024,1702,2768,4422,6948,10752,16424,
%T A022569 24782,36972,54602,79872,115805,166540,237664,336720,473856,662596,
%U A022569 920934,1272728,1749407,2392268,3255410,4409344,5945730,7983388,10675712,14220240,18870672,24951740,32878114
%N A022569 Expansion of Product_{m>=1} (1+x^m)^4.
%C A022569 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A022569 Seiichi Manyama, <a href="/A022569/b022569.txt">Table of n, a(n) for n = 0..1000</a>
%H A022569 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 A022569 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A022569 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A022569 Expansion of q^(-1/6) * (eta(q^2) / eta(q))^4 in powers of q.
%F A022569 Expansion of chi(-q)^(-4) in powers of q where chi() is a Ramanujan theta function.
%F A022569 Euler transform of period 2 sequence [ 4, 0, ...]. - _Michael Somos_, Apr 26 2008
%F A022569 Given G.f. A(x) then B(q) = (A(q^6) * q)^2 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v * (1 + 16 * u * v) - u^2. - _Michael Somos_, Apr 26 2008
%F A022569 Given G.f. A(x) then B(x) = A(q^6) * q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v * (u^2 - v) - 4 * w^2 * (u^2 + v). - _Michael Somos_, Apr 26 2008
%F A022569 G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A022599.
%F A022569 G.f.: Product_{k>0} (1 + x^k)^4.
%F A022569 Convolution inverse of A022599.
%F A022569 G.f.: T(0)/x, where T(k) = 1 - 1/(1 - (1+(x)^(k+1))^4/((1+(x)^(k+1))^4 - 1/T(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Nov 07 2013
%F A022569 a(n) ~ exp(2 * Pi * sqrt(n/3)) / (8 * 3^(1/4) * n^(3/4)) * (1 + (Pi/(6*sqrt(3)) - 3*sqrt(3)/(16*Pi)) / sqrt(n)). - _Vaclav Kotesovec_, Mar 05 2015, extended Jan 16 2017
%F A022569 a(0) = 1, a(n) = (4/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 03 2017
%F A022569 G.f.: exp(4*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 06 2018
%e A022569 G.f. = 1 + 4*x + 10*x^2 + 24*x^3 + 51*x^4 + 100*x^5 + 190*x^6 + 344*x^7 + ...
%e A022569 G.f. = q + 4*q^7 + 10*q^13 + 24*q^19 + 51*q^25 + 100*q^31 + 190*q^37 + 344*q^43 + ...
%t A022569 a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^-4, {q, 0, n}]; (* _Michael Somos_, Jul 11 2011 *)
%t A022569 a[ n_] := SeriesCoefficient[ Product[ 1 + q^k, {k, n}]^4, {q, 0, n}]; (* _Michael Somos_, Jul 11 2011 *)
%o A022569 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) / eta(x + A))^4, n))}; /* _Michael Somos_, Apr 26 2008 */
%o A022569 (PARI) m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^4)) \\ _G. C. Greubel_, Feb 26 2018
%o A022569 (Magma) Coefficients(&*[(1+x^m)^4:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // _G. C. Greubel_, Feb 26 2018
%Y A022569 Cf. A000009, A022599.
%Y A022569 Column k=4 of A286335.
%K A022569 nonn
%O A022569 0,2
%A A022569 _N. J. A. Sloane_