cp's OEIS Frontend

A259910 Expansion of f(-x^2, -x^3)^3 / f(-x)^2 in powers of x where f(,) is the Ramanujan general theta function.

%I A259910 #22 Feb 16 2025 08:33:26
%S A259910 1,2,2,1,2,3,4,3,3,4,6,6,7,8,8,9,11,12,13,14,17,19,21,21,25,27,30,31,
%T A259910 35,39,43,47,51,55,60,65,71,77,83,88,98,105,115,122,132,142,155,164,
%U A259910 178,191,206,220,236,252,272,290,311,332,356,378,407,434,464
%N A259910 Expansion of f(-x^2, -x^3)^3 / f(-x)^2 in powers of x where f(,) is the Ramanujan general theta function.
%C A259910 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A259910 Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).
%D A259910 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, 6th equation.
%H A259910 Vaclav Kotesovec, <a href="/A259910/b259910.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)
%H A259910 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A259910 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A259910 Expansion of f(-x) * (f(-x^5) / f(-x, -x^4))^3 in powers of x where f(,) is the Ramanujan general theta function.
%F A259910 Expansion of f(-x^2, -x^3) * G(x)^2 in powers of x where f(,) is the Ramanujan general theta function and G() is a Rogers-Ramanujan function. - _Michael Somos_, Jul 09 2015
%F A259910 Euler transform of period 5 sequence [ 2, -1, -1, 2, -1, ...].
%F A259910 G.f.: (Sum_{k in Z} (-1)^k * x^(k*(5*k + 1)/2)) / (Product_{k in Z} 1 - x^abs(5*k + 1))^2.
%F A259910 a(n) = 3 * A053266(n) - A053262(n) unless n=0.
%F A259910 a(n) ~ sqrt(5+2*sqrt(5)) * exp(sqrt(2*n/15)*Pi)/ (5*sqrt(2*n)). - _Vaclav Kotesovec_, Dec 17 2016
%e A259910 G.f. = 1 + 2*x + 2*x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 3*x^7 + 3*x^8 + ...
%e A259910 G.f. = 1/q + 2*q^119 + 2*q^239 + q^359 + 2*q^479 + 3*q^599 + 4*q^719 + ...
%t A259910 a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ -2, 1, 1, -2, 1}[[
%t A259910 Mod[k, 5, 1]]], {k, n}], {x, 0, n}];
%t A259910 nmax = 100; CoefficientList[Series[Product[(1-x^k)/((1 - x^(5*k-1))*(1 - x^(5*k-4)))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Dec 17 2016 *)
%o A259910 (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, -2, 1, 1, -2][k%5+1]), n))};
%Y A259910 Cf. A053262, A053266.
%K A259910 nonn
%O A259910 0,2
%A A259910 _Michael Somos_, Jul 07 2015