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A239051 Expansion of (f(-q^2, -q^3)^5 - 3 * q * f(-q, -q^4)^5) / f(-q)^3 in powers of q where f() is a Ramanujan theta function.

%I A239051 #20 Jun 14 2025 01:50:07
%S A239051 1,0,10,-10,10,0,0,10,0,-10,10,0,10,-10,20,-10,0,10,-10,0,10,0,20,-10,
%T A239051 0,0,0,0,10,0,0,0,10,-20,20,10,0,10,0,-20,0,0,20,-10,20,-10,0,10,-10,
%U A239051 10,10,0,10,-10,0,0,0,0,0,0,10,0,20,-10,10,-10,0,10,10
%N A239051 Expansion of (f(-q^2, -q^3)^5 - 3 * q * f(-q, -q^4)^5) / f(-q)^3 in powers of q where f() is a Ramanujan theta function.
%C A239051 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A239051 G. C. Greubel, <a href="/A239051/b239051.txt">Table of n, a(n) for n = 0..1000</a>
%H A239051 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A239051 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A239051 Moebius transform is period 5 sequence [ 0, 10, -10, 0, 0, ...].
%F A239051 G.f.: 1 + 10 * ( Sum_{k>=0} x^(5*k + 2) / (1 - x^(5*k + 2)) - x^(5*k + 3) / (1 - x^(5*k + 3)) ).
%F A239051 a(n) = A227216(n) - 3 * A229802(n).
%F A239051 a(5*n) = a(n). a(5*n + 1) = 0.
%e A239051 G.f. = 1 + 10*q^2 - 10*q^3 + 10*q^4 + 10*q^7 - 10*q^9 + 10*q^10 + 10*q^12 + ...
%t A239051 a[ n_] := If[ n < 1, Boole[ n == 0], 10 Sum[ {0, 1, -1, 0, 0}[[ Mod[ d, 5, 1] ]], {d, Divisors @ n}]];
%o A239051 (PARI) {a(n) = if( n<1, n==0, 10 * sumdiv(n, d, (d%5==2) - (d%5==3)))};
%o A239051 (Sage) ModularForms( Gamma1(5), 1, prec=70).0;
%o A239051 (Magma) Basis( ModularForms( Gamma1(5), 1), 70) [1];
%Y A239051 Cf. A227216, A229802.
%K A239051 sign
%O A239051 0,3
%A A239051 _Michael Somos_, Jun 13 2014