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URL: https://oeis.org/A288143

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A288143
Expansion of x * phi(x) * phi(x^3)^2 * f(x, x^5)^3 in powers of x where phi() is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.
2
1, 5, 9, 11, 24, 45, 50, 53, 81, 120, 120, 99, 170, 250, 216, 203, 288, 405, 362, 264, 450, 600, 528, 477, 601, 850, 729, 550, 840, 1080, 962, 821, 1080, 1440, 1200, 891, 1370, 1810, 1530, 1272, 1680, 2250, 1850, 1320, 1944, 2640, 2208, 1827, 2451, 3005, 2592
OFFSET
1,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
Expansion of (a(q^2) - a(-q)) * (2*a(q) + a(-q))^2 / 54 in powers of q where a() is a cubic AGM theta function.
Expansion of -c(-q) * (2*c(q) + c(-q))^2 / 27 in powers of q where c() is a cubic AGM theta function.
Expansion of eta(q^2)^11 * eta(q^6)^7 / (eta(q)^5 * eta(q^3) * eta(q^4)^5 * eta(q^12)) in powers of q.
a(n) is multiplicative with a(3^e) = 9^e, a(2^e) = (4^(e+1) + 9*(-1)^(e+1)) / 5 if e>0, a(p^e) = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1 (mod 6), a(p^e) = ((p^2)^(e+1) - (-1)^(e+1)) / (p^2 + 1) if p == 5 (mod 6).
Euler transform of period 12 sequence [5, -6, 6, -1, 5, -12, 5, -1, 6, -6, 5, -6, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 192^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113261.
G.f.: Sum_{k>0} k^2 * x^k / (1 + x^k + x^(2*k)) * if(mod(k,4)=2, 3/2, 1).
a(n) = -(-1)^n * A214262(n).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Product_{p prime == 1 (mod 6)} (p^3/(p^3-1)) * Product_{p prime == 5 (mod 6)} (p^3/(p^3+1)) = 1/(A334478 * A334479) = 0.99452678821883983883... . - Amiram Eldar, Feb 20 2024
Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/54) * exp(Pi) * 3^(3/4) * Pi^2 * (3^(1/2)-1) / Gamma(2/3) / Gamma(3/4)^5 / Gamma(7/12) = A389025. - Simon Plouffe, Sep 22 2025
EXAMPLE
G.f. = q + 5*q^2 + 9*q^3 + 11*q^4 + 24*q^5 + 45*q^6 + 50*q^7 + 53*q^8 + 81*q^9 + ...
MATHEMATICA
a[ n_] := If[ n < 1, 0, (-1)^n DivisorSum[ n, (-1)^# #^2 JacobiSymbol[ -3, n/#] &]];
a[ n_] := SeriesCoefficient[ x EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^3]^2 (QPochhammer[ -x, x^6] QPochhammer[ -x^5, x^6] QPochhammer[ x^6])^3, {x, 0, n}];
a[ n_] := If[ n < 2, Boole[n == 1], Times @@ (Which[# == 3, 9^#2, # == 2, (4^(#2 + 1) + 9 (-1)^(#2 + 1))/5, Mod[#, 6] == 1, ((#^2)^(#2 + 1) - 1)/(#^2 - 1), True, ((#^2)^(#2 + 1) - (-1)^(#2 + 1))/(#^2 + 1)] & @@@ FactorInteger@n)];
PROG
(PARI) {a(n) = if( n<1, 0, (-1)^n * sumdiv( n, d, (-1)^d * d^2 * kronecker( -3, n/d)))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^11 * eta(x^6 + A)^7 / (eta(x + A)^5 * eta(x^3 + A) * eta(x^4 + A)^5 * eta(x^12 + A)), n))};
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 9^e, p==2, (4^(e+1) + 9*(-1)^(e+1)) / 5, p%6==1, ((p^2)^(e+1) - 1) / (p^2 - 1), ((p^2)^(e+1) - (-1)^(e+1)) / (p^2 + 1))))};
(Magma) A := Basis( ModularForms( Gamma1(12), 3), 52); A[2] + 5*A[3] + 9*A[4] + 11*A[5] + 24*A[6] + 45*A[7] + 50*A[8] + 53*A[9] + 81*A[10] + 120*A[11] + 120*A[12] + 99*A[13];
KEYWORD
nonn,mult
AUTHOR
Michael Somos, Jul 01 2017
STATUS
approved