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.

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

Indranil Ghosh, Table of n, a(n) for n = 1..2000

Michael Somos, Introduction to Ramanujan theta functions, 2019.

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

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];

Crossrefs

Cf. A113261, A214262, A334478, A334479.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

Sequence in context: A309137 A190894 A214262 * A120228 A053749 A099082

Adjacent sequences: A288140 A288141 A288142 * A288144 A288145 A288146

Keyword

nonn,mult

Author

Michael Somos, Jul 01 2017

Status

approved