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A263002 Expansion of (f(-x^5) / f(-x))^2 in powers of x where f() is a Ramanujan theta function.

%I A263002 #15 Feb 16 2025 08:33:27
%S A263002 1,2,5,10,20,34,61,100,165,260,408,620,940,1390,2045,2960,4257,6040,
%T A263002 8525,11900,16522,22738,31130,42300,57210,76872,102834,136800,181230,
%U A263002 238900,313725,410160,534330,693330,896655,1155420,1484274,1900420,2426215,3088100
%N A263002 Expansion of (f(-x^5) / f(-x))^2 in powers of x where f() is a Ramanujan theta function.
%C A263002 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A263002 Number of 5-regular bipartitions of n. - _N. J. A. Sloane_, Oct 20 2019
%D A263002 Kathiravan, T., and S. N. Fathima. "On L-regular bipartitions modulo L." The Ramanujan Journal 44.3 (2017): 549-558.
%H A263002 Vaclav Kotesovec, <a href="/A263002/b263002.txt">Table of n, a(n) for n = 0..1000</a>
%H A263002 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
%H A263002 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A263002 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A263002 Expansion of q^(-1/3) * (eta(q^5) / eta(q))^2 in powers of q.
%F A263002 Euler transform of period 5 sequence [ 2, 2, 2, 2, 0, ...].
%F A263002 G.f. is a period 1 Fourier series which satisfies f(-1 / (45 t)) = (1/5) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A058511.
%F A263002 Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u - v^2) * (v - u^2) - 4*u^2*v^2.
%F A263002 Convolution inverse is A058511.
%F A263002 a(n) ~ exp(4*Pi*sqrt(n/15)) / (sqrt(2) * 3^(1/4) * 5^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 14 2015
%F A263002 See Maple code for a simple g.f. - _N. J. A. Sloane_, Oct 20 2019
%e A263002 G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 34*x^5 + 61*x^6 + 100*x^7 + ...
%e A263002 G.f. = q + 2*q^4 + 5*q^7 + 10*q^10 + 20*q^13 + 34*q^16 + 61*q^19 + 100*q^22 + ...
%p A263002 f:=(k,M) -> mul(1-q^(k*j),j=1..M);
%p A263002 LRBP := (L,M) -> (f(L,M)/f(1,M))^2;
%p A263002 S := L -> seriestolist(series(LRBP(L,80),q,60));
%p A263002 S(5); # _N. J. A. Sloane_, Oct 20 2019
%t A263002 a[ n_] := SeriesCoefficient[ (QPochhammer[ x^5] / QPochhammer[ x])^2, {x, 0, n}];
%o A263002 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^5 + A) / eta(x + A))^2, n))};
%Y A263002 Cf. A058511.
%Y A263002 Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548.
%K A263002 nonn
%O A263002 0,2
%A A263002 _Michael Somos_, Oct 07 2015