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A229802 Expansion of q * f(-q, -q^4)^5 / f(-q)^3 in powers of q where f() is a Ramanujan theta function.

%I A229802 #22 Feb 16 2025 08:33:20
%S A229802 1,-2,4,-3,1,2,-2,0,3,-2,2,-2,4,-6,4,1,-2,4,0,-3,2,-4,4,0,1,2,0,-4,0,
%T A229802 2,2,-2,8,-6,-2,1,-2,0,6,0,2,-4,4,-6,3,2,-2,4,-3,-2,2,-2,4,0,2,0,0,0,
%U A229802 0,-2,2,-4,4,-3,4,4,-2,-4,6,-6,2,0,4,-6,4,0,-4
%N A229802 Expansion of q * f(-q, -q^4)^5 / f(-q)^3 in powers of q where f() is a Ramanujan theta function.
%C A229802 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A229802 G. C. Greubel, <a href="/A229802/b229802.txt">Table of n, a(n) for n = 1..1000</a>
%H A229802 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A229802 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A229802 Expansion of q * f(-q)^2 * (f(-q^5) / f(-q^2, -q^3))^5 = q * f(-q, -q^4)^2 * (f(-q^5) / f(-q^2, -q^3))^3 in powers of q where f() is a Ramanujan theta function. - _Michael Somos_, Jun 10 2014
%F A229802 Euler transform of period 5 sequence [ -2, 3, 3, -2, -2, ...].
%F A229802 Moebius transform is period 5 sequence [ 1, -3, 3, -1, 0, ...]. - _Michael Somos_, Jun 10 2014
%F A229802 G.f.: x * (Product_{k>0} (1 - x^k)^2) / (Product_{k>0} (1 - x^(5*k - 2)) * (1 - x^(5*k - 3)))^5.
%F A229802 a(5*n) = a(n).
%e A229802 G.f. = q - 2*q^2 + 4*q^3 - 3*q^4 + q^5 + 2*q^6 - 2*q^7 + 3*q^9 - 2*q^10 + ...
%t A229802 a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^2 / (QPochhammer[ q^2, q^5] QPochhammer[ q^3, q^5])^5, {q, 0, n}]; (* _Michael Somos_, Jun 10 2014 *)
%t A229802 a[ n_] := If[ n < 1, 0, Sum[ Im[(I - 3) {1, I, -I, -1, 0}[[ Mod[ d, 5, 1] ]] ], {d, Divisors @ n}]]; (* _Michael Somos_, Jun 10 2014 *)
%o A229802 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( prod(k=1, n, (1 - x^k + A)^[ 2, 2, -3, -3, 2][k%5 + 1]), n))};
%o A229802 (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, imag( (I - 3) * [ 0, 1, I, -I, -1][ d%5 + 1])))}; /* _Michael Somos_, Jun 10 2014 */
%o A229802 (Sage) ModularForms( Gamma1(5), 1, prec=70).1;
%o A229802 (Magma) Basis( ModularForms( Gamma1(5), 1), 70) [2] ; /* _Michael Somos_, Jun 10 2014 */
%Y A229802 Cf. A227216.
%K A229802 sign
%O A229802 1,2
%A A229802 _Michael Somos_, Sep 29 2013