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%I #20 Nov 24 2023 08:13:35
%S 1,3,2,1,5,6,6,7,7,15,12,2,12,18,10,9,16,21,20,5,12,36,22,14,25,36,20,
%T 6,30,30,32,23,24,48,30,7,36,60,24,35,42,36,42,12,35,66,46,18,43,75,
%U 32,12,52,60,60,42,40,90,60,10,62,96,42,41,60,72,66,16,44
%N Expansion of (phi(q)^3 * phi(q^5) - phi(q) * phi(q^5)^3) / 4 in powers of q where phi() is a Ramanujan theta function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A138483/b138483.txt">Table of n, a(n) for n = 1..10000</a>
%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 * phi(q) * chi(q) * f(q^5) * f(-q^10)^2 in powers of q where phi(), chi(), f() are Ramanujan theta functions.
%F Expansion of eta(q^2)^7 * eta(q^10)^5 / (eta(q)^3 * eta(q^4)^3 * eta(q^5) * eta(q^20)) in powers of q.
%F Euler transform of period 20 sequence [3, -4, 3, -1, 4, -4, 3, -1, 3, -8, 3, -1, 3, -4, 4, -1, 3, -4, 3, -4, ...].
%F a(n) is multiplicative with a(2^e) = (2^(e+1) - 5*(-1)^e) / 3 if e>0, a(5^e) = 5^e, a(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 4 (mod 5), a(p^e) = (p^(e+1) + (-1)^e) / (p + 1) if p == 2, 3 (mod 5).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 80^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113185.
%F G.f.: x * Product_{k>0} (1 - x^(2*k)) * (1 + x^(2*k-1))^3 * (1 - x^(5*k))^3 * (1 + x^(10*k-5))^4 * (1 + x^(10*k))^3.
%F a(n) = -(-1)^n * A110712(n).
%F Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Pi^2/(5*sqrt(5)) = 0.882764... . - _Amiram Eldar_, Nov 24 2023
%e G.f. = q + 3*q^2 + 2*q^3 + q^4 + 5*q^5 + 6*q^6 + 6*q^7 + 7*q^8 + 7*q^9 + ...
%t a[ n_] := If[ n < 1, 0, DivisorSum[ n, n/# KroneckerSymbol[ 25, #] (-1)^Quotient[# + 2, 5] If[ Mod[#, 4] > 0, 1, 5] &]]; (* _Michael Somos_, Sep 27 2015 *)
%t a[ n_] := SeriesCoefficient[ q EllipticTheta[3, 0, q] QPochhammer[ -q, q^2] QPochhammer[ -q^5] QPochhammer[ q^10]^2, {q, 0, n}]; (* _Michael Somos_, Sep 27 2015 *)
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[3, 0, q]^3 EllipticTheta[3, 0, q^5] - EllipticTheta[3, 0, q] EllipticTheta[3, 0, q^5]^3) / 4, {q, 0, n}]; (* _Michael Somos_, Sep 27 2015 *)
%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, n/d * kronecker(25, d) * (-1)^((d+2) \ 5) * if(d%4, 1, 5)))};
%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, ]; if( p==2, (2^(e+1) - 5*(-1)^e) / 3, f = kronecker(5, 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)^7 * eta(x^10 + A)^5 / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^5 + A) * eta(x^20 + A)), n))};
%Y Cf. A110712, A113185.
%Y Cf. A000122, A000700, A010054, A121373.
%K nonn,easy,mult
%O 1,2
%A _Michael Somos_, Mar 20 2008
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