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A124233 Expansion of psi(q) * phi(-q^10) * chi(-q^5) / chi(-q^2) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

%I A124233 #32 Aug 06 2025 10:13:18
%S A124233 1,1,1,2,1,1,2,2,1,3,1,0,2,0,2,2,1,0,3,0,1,4,0,2,2,1,0,4,2,2,2,0,1,0,
%T A124233 0,2,3,0,0,0,1,2,4,2,0,3,2,2,2,3,1,0,0,0,4,0,2,0,2,0,2,2,0,6,1,0,0,2,
%U A124233 0,4,2,0,3,0,0,2,0,0,0,0,1,5,2,2,4,0,2
%N A124233 Expansion of psi(q) * phi(-q^10) * chi(-q^5) / chi(-q^2) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
%C A124233 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A124233 Number 38 of the 74 eta-quotients listed in Table I of Martin (1996).
%H A124233 G. C. Greubel, <a href="/A124233/b124233.txt">Table of n, a(n) for n = 0..10000</a>
%H A124233 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 A124233 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 A124233 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 A124233 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A124233 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A124233 Expansion of eta(q^2) * eta(q^4) * eta(q^5) * eta(q^10) / (eta(q) * eta(q^20)) in powers of q.
%F A124233 Euler transform of period 20 sequence [ 1, 0, 1, -1, 0, 0, 1, -1, 1, -2, 1, -1, 1, 0, 0, -1, 1, 0, 1, -2, ...].
%F A124233 Moebius transform is period 20 sequence [ 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, ...].
%F A124233 a(n) is multiplicative with a(2^e) = a(5^e) = 1, a(p^e) = e+1 if p == 1, 3, 7, 9 (mod 20), a(p^e) = (1 + (-1)^e) / 2 if p == 11, 13, 17, 19 (mod 20).
%F A124233 G.f.: 1 + Sum_{k>0} x^k * (1 + x^(2*k)) * (1 + x^(6*k)) / (1 + x^(10*k)).
%F A124233 a(2*n) = a(5*n) = a(n), a(20*n + 11) = a(20*n + 13) = a(20*n + 17) = a(20*n + 19) = 0.
%F A124233 a(n) = A035170(n) unless n=0. a(2*n + 1) = A129390(n). a(4*n + 3) = 2 * A033764(n).
%F A124233 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(5) = 1.404962... . - _Amiram Eldar_, Dec 22 2023
%e A124233 G.f. = 1 + q + q^2 + 2*q^3 + q^4 + q^5 + 2*q^6 + 2*q^7 + q^8 + 3*q^9 + q^10 + ...
%t A124233 a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, KroneckerSymbol[ -20, #] &]]; (* _Michael Somos_, Jul 09 2015 *)
%t A124233 a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^4] QPochhammer[ q^5] QPochhammer[ q^10] / (QPochhammer[ q] QPochhammer[ q^20]), {q, 0, n}]; (* _Michael Somos_, Jul 09 2015 *)
%o A124233 (PARI) {a(n) = if( n<1, n==0, sumdiv( n, d, kronecker( -20, d)))};
%o A124233 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^10 + A) / eta(x + A) / eta(x^20 + A), n))};
%Y A124233 Cf. A033764, A035170, A129390.
%Y A124233 Cf. A000122, A000700, A010054, A121373.
%K A124233 nonn,easy,mult
%O A124233 0,4
%A A124233 _Michael Somos_, Oct 21 2006