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A128616 Expansion of q * psi(q^3) * psi(q^5) in powers of q where psi() is a Ramanujan theta function.

%I A128616 #13 Feb 16 2025 08:33:05
%S A128616 1,0,0,1,0,1,0,0,1,1,0,0,0,0,1,1,0,0,2,0,0,0,0,1,1,0,0,0,0,0,2,0,0,2,
%T A128616 0,1,0,0,0,1,0,0,0,0,0,2,0,0,1,0,2,0,0,1,0,0,0,0,0,1,2,0,0,1,0,0,0,0,
%U A128616 2,0,0,0,0,0,0,2,0,0,2,0,1,0,0,0,2,0,0
%N A128616 Expansion of q * psi(q^3) * psi(q^5) in powers of q where psi() is a Ramanujan theta function.
%C A128616 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A128616 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 377, Entry 9(iv).
%H A128616 G. C. Greubel, <a href="/A128616/b128616.txt">Table of n, a(n) for n = 1..10000</a>
%H A128616 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A128616 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A128616 Expansion of (eta(q^6) * eta(q^10))^2 / (eta(q^3) * eta(q^5)) in powers of q.
%F A128616 Euler transform of period 30 sequence [ 0, 0, 1, 0, 1, -1, 0, 0, 1, -1, 0, -1, 0, 0, 2, 0, 0, -1, 0, -1, 1, 0, 0, -1, 1, 0, 1, 0, 0, -2, ...].
%F A128616 For n>0, n in A028957 equivalent to a(n) nonzero. If a(n) nonzero, a(n) = A082451(n) and a(n) = A121362(n).
%F A128616 a(n) = (A082451(n) + A121362(n))/2.
%F A128616 G.f.: x * Product_{k>0} (1 - x^(3*k)) * (1 - x^(5*k)) * (1 + x^(6*k))^2 * (1 + x^(10*k))^2.
%e A128616 G.f. = x + x^4 + x^6 + x^9 + x^10 + x^15 + x^16 + 2*x^19 + x^24 + x^25 + 2*x^31 + ...
%t A128616 a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -60, #] + KroneckerSymbol[ 20, #] KroneckerSymbol[ -3, n/#] &] / 2]; (* _Michael Somos_, Nov 12 2015 *)
%t A128616 a[ n_] := SeriesCoefficient[ q(QPochhammer[ q^6] QPochhammer[ q^10])^2 / (QPochhammer[ q^3] QPochhammer[ q^5]), {q, 0, n}]; (* _Michael Somos_, Nov 12 2015 *)
%o A128616 (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-60, d) + kronecker(20, d) * kronecker(-3, n/d) )/2)};
%o A128616 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^6 + A) * eta(x^10 + A))^2 / (eta(x^3 + A) * eta(x^5 + A)), n))};
%Y A128616 Cf. A082451, A121362.
%K A128616 nonn
%O A128616 1,19
%A A128616 _Michael Somos_, Mar 13 2007