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A145933 Expansion of chi(-q^6) * chi(-q^10) / (chi(-q) * chi(-q^15)) in powers of q where chi() is a Ramanujan theta function.

%I A145933 #19 Feb 16 2025 08:33:09
%S A145933 1,1,1,2,2,3,3,4,5,6,7,8,10,11,14,17,20,23,27,31,36,41,48,55,63,72,82,
%T A145933 94,106,122,137,156,175,197,222,249,280,314,352,393,439,490,546,608,
%U A145933 676,751,834,923,1024,1133,1253,1384,1528,1686,1857,2045,2250,2474,2718,2984
%N A145933 Expansion of chi(-q^6) * chi(-q^10) / (chi(-q) * chi(-q^15)) in powers of q where chi() is a Ramanujan theta function.
%C A145933 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A145933 Seiichi Manyama, <a href="/A145933/b145933.txt">Table of n, a(n) for n = 0..10000</a>
%H A145933 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A145933 Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/retaprod.html">A Remarkable eta-product Identity</a>
%H A145933 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A145933 Expansion of eta(q^2) * eta(q^6) * eta(q^10) * eta(q^30) / (eta(q) * eta(q^12) * eta(q^15) * eta(q^20)) in powers of q.
%F A145933 Euler transform of a period 60 sequence.
%F A145933 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 2 * (1 - u)^2 * (1 - v)^2 + 2 * u*v * (u - v)^2 + u*v * (v - u^2) * (u - v^2).
%F A145933 G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A058728.
%F A145933 G.f.: Product_{k>0} (1 + x^k) * (1 + x^(15*k)) / ((1 + x^(6*k)) * (1 + x^(10*k))).
%F A145933 a(n) = A143752(n) unless n=0. a(2*n) = A094203(n).
%F A145933 Convolution inverse of A132968.
%F A145933 a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 15^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 08 2015
%e A145933 G.f. = 1 + q + q^2 + 2*q^3 + 2*q^4 + 3*q^5 + 3*q^6 + 4*q^7 + 5*q^8 + 6*q^9 + ...
%t A145933 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ -x^15, x^15] / (QPochhammer[ -x^6, x^6] QPochhammer[ -x^10, x^10]), {x, 0, n}]; (* _Michael Somos_, Sep 03 2015 *)
%t A145933 nmax = 60; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(15*k)) / ((1 + x^(6*k)) * (1 + x^(10*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 08 2015 *)
%o A145933 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^30 + A) / (eta(x + A) * eta(x^12 + A) * eta(x^15 + A) * eta(x^20 + A)), n))};
%Y A145933 Cf. A058728, A094203, A132968, A143752.
%K A145933 nonn
%O A145933 0,4
%A A145933 _Michael Somos_, Oct 25 2008