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A124815 Expansion of q * psi(q)^2 * psi(-q^3)^2 * phi(-q^6) / phi(-q^2) in powers of q where phi(), psi() are Ramanujan theta functions.

%I A124815 #33 Aug 06 2025 10:08:34
%S A124815 1,2,3,4,4,6,6,8,9,8,12,12,14,12,12,16,16,18,18,16,18,24,24,24,21,28,
%T A124815 27,24,28,24,30,32,36,32,24,36,38,36,42,32,40,36,42,48,36,48,48,48,43,
%U A124815 42,48,56,52,54,48,48,54,56,60,48,62,60,54,64,56,72,66,64,72,48,72,72
%N A124815 Expansion of q * psi(q)^2 * psi(-q^3)^2 * phi(-q^6) / phi(-q^2) in powers of q where phi(), psi() are Ramanujan theta functions.
%C A124815 Number 24 of the 74 eta-quotients listed in Table I of Martin (1996).
%C A124815 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A124815 G. C. Greubel, <a href="/A124815/b124815.txt">Table of n, a(n) for n = 1..5000</a>
%H A124815 David Broadhurst and Daniele Dorigoni, <a href="https://arxiv.org/abs/2507.21352">Resurgent Lambert series with characters</a>, arXiv:2507.21352 [math.NT], 2025. See p. 37.
%H A124815 Yves Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852, Table I.
%H A124815 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>, 2016.
%H A124815 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A124815 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A124815 Expansion of (eta(q^2) * eta(q^3) / eta(q))^2 * eta(q^4) * eta(q^12) in powers of q.
%F A124815 Euler transform of period 12 sequence [ 2, 0, 0, -1, 2, -2, 2, -1, 0, 0, 2, -4, ...].
%F A124815 a(n) is multiplicative with a(p^e) = p^e if p<5, a(p^e) = (p^(e+1) - 1) / (p-1) if p == 1, 11 (mod 12), a(p^e) = (p^(e+1) + (-1)^e) / (p+1) if p == 5, 7 (mod 12).
%F A124815 G.f.: Sum_{k>0} k * x^k * (1 - x^(2*k)) / (1 - x^(2*k) + x^(4*k)).
%F A124815 G.f.: x * Product_{k>0} (1 + x^k)^2 * (1 - x^(3*k))^2 * (1 - x^(4*k)) * (1 - x^(12*k)).
%F A124815 a(2*n) = 2 * a(n).
%F A124815 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Pi^2/(6*sqrt(3)) = 0.949703... (A258414). - _Amiram Eldar_, Dec 22 2023
%e A124815 G.f. = q + 2*q^2 + 3*q^3 + 4*q^4 + 4*q^5 + 6*q^6 + 6*q^7 + 8*q^8 + 9*q^9 + ...
%t A124815 a[ n_] := If[ n < 1, 0, Sum[ n/d KroneckerSymbol[ 12, d], { d, Divisors[ n]}]]; (* _Michael Somos_, Jul 09 2015 *)
%t A124815 a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2]^2 QPochhammer[ q^3]^2 QPochhammer[ q^4] QPochhammer[ q^12]/QPochhammer[ q]^2, {q, 0, n}]; (* _Michael Somos_, Jul 09 2015 *)
%o A124815 (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, n/d * kronecker( 12, d)))};
%o A124815 (PARI) {a(n) = my(A, p, e, f); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; f = kronecker( 12, p); (p^(e+1) - f^(e+1)) / (p - f)))};
%o A124815 (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^12 + A) / eta(x + A)^2, n))};
%Y A124815 Cf. A258414.
%Y A124815 Cf. A000122, A000700, A010054, A121373.
%K A124815 nonn,easy,mult
%O A124815 1,2
%A A124815 _Michael Somos_, Nov 08 2006