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%I #27 Dec 22 2023 08:29:31
%S 1,2,2,4,5,4,6,8,7,10,12,8,12,12,10,16,16,14,20,20,12,24,22,16,25,24,
%T 20,24,30,20,32,32,24,32,30,28,36,40,24,40,42,24,42,48,35,44,46,32,43,
%U 50,32,48,52,40,60,48,40,60,60,40,62,64,42,64,60,48,66,64
%N Expansion of eta(q^2)^3 * eta(q^5)^2 * eta(q^10) / eta(q)^2 in powers of q.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H Vaclav Kotesovec, <a href="/A129303/b129303.txt">Table of n, a(n) for n = 1..1000</a>
%H Shaun Cooper, <a href="https://doi.org/10.1007/s11139-009-9198-5">On Ramanujan's function k(q)=r(q)r^2(q^2)</a>, Ramanujan J., 20 (2009), 311-328; <a href="https://www.researchgate.net/publication/265424799_On_Ramanujan%27s_function_kqrqr_2_q_2">ResearchGate link</a>. See p. 318, Th. 4.1.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F Expansion of q * psi(q)^3 * psi(q^5) - q^2 * psi(q) * psi(q^5)^3 in powers of q where psi() is a Ramanujan theta function. - _Michael Somos_, Jul 12 2012
%F Euler transform of period 10 sequence [ 2, -1, 2, -1, 0, -1, 2, -1, 2, -4, ...].
%F a(n) is multiplicative with a(p^e) = p^e if p = 2 or 5, a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 9 (mod 10), a(p^e) = (p^(e+1) + (-1)^e) / (p + 1) if p == 3, 7 (mod 10).
%F G.f.: Sum_{k>0} Kronecker(20, k) * x^k / (1 - x^k)^2.
%F G.f.: x * Product_{k>0} (1 - x^k) * (1 + x^(5*k)) * (1 + x^k)^3 * (1 - x^(5*k))^3.
%F a(2*n) = a(n). a(2*n + 1) = A134080(n).
%F Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Pi^2/(5*sqrt(5)) = 0.882764... . - _Amiram Eldar_, Dec 22 2023
%e G.f. = q + 2*q^2 + 2*q^3 + 4*q^4 + 5*q^5 + 4*q^6 + 6*q^7 + 8*q^8 + 7*q^9 + ...
%t a[ n_] := If[ n < 1, 0, DivisorSum[ n, n/# KroneckerSymbol[ 20, #] &]]; (* _Michael Somos_, Jul 12 2012 *)
%t a[ n_] := SeriesCoefficient[ (1/16) (EllipticTheta[ 2, 0, q]^3 EllipticTheta[ 2, 0, q^5] - EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^5]^3), {q, 0, 2 n}]; (* _Michael Somos_, Jul 12 2012 *)
%t nmax = 100; Rest[CoefficientList[Series[x * Product[(1 - x^k) * (1 + x^(5*k)) * (1 + x^k)^3 * (1 - x^(5*k))^3, {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Sep 08 2015 *)
%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, n/d * kronecker( 20, d)))};
%o (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( 20, p); (p^(e+1) - f^(e+1)) / (p - f) ))};
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^5 + A)^2 * eta(x^10 + A) / eta(x + A)^2, n))};
%Y Cf. A134080.
%Y Cf. A000122, A000700, A010054, A121373.
%K nonn,easy,mult
%O 1,2
%A _Michael Somos_, Apr 08 2007
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