OFFSET
0,6
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
Expansion of eta(q^2) * eta(q^3)^2 * eta(q^12)^2 * eta(q^18) / (eta(q) * eta(q^4) * eta(q^9) * eta(q^36)) in powers of q.
Euler transform of period 36 sequence [1, 0, -1, 1, 1, -2, 1, 1, 0, 0, 1, -3, 1, 0, -1, 1, 1, -2, 1, 1, -1, 0, 1, -3, 1, 0, 0, 1, 1, -2, 1, 1, -1, 0, 1, -2, ...].
Moebius transform is period 36 sequence [1, 0, -1, 0, 1, 0, -1, 0, 4, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -4, 0, 1, 0, -1, 0, 1, 0, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) f(t) where q = exp(2 Pi i t).
a(2*n) = a(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). a(4*n + 3) = 0. a(12*n + 1) = A002175. a(12*n + 5) = 2 * A121444(n).
a(n) = Sum_{d|n} A258260(d) * (-1)^(n+d) if n>0.
a(n) = (-1)^n * A256282(n). - Michael Somos, Jun 06 2015
a(n) is multiplicative with a(0) = 1, a(2^e) = 1, a(3^e) = 2*(1 + (-1)^e), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), a(p^e) = e+1 if p == 1 (mod 4). - Michael Somos, Jun 06 2015
Expansion of A0(x)^2 + A0(x)*A1(x) + A1(x)^2 in powers of x where A0(x) = phi(x^9), A1(x) = x * f(x^3, x^15) = x * A089801(x^3). - Michael Somos, Jun 23 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 (A019670). - Amiram Eldar, Nov 24 2023
EXAMPLE
G.f. = 1 + q + q^2 + q^4 + 2*q^5 + q^8 + 4*q^9 + 2*q^10 + 2*q^13 + q^16 + ...
MATHEMATICA
a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, {1, 2, -1, 0}[[Mod[#, 4, 1]]] If[ Divisible[ #, 9], 4, 1] (-1)^(Boole[Mod[#, 8] == 6] + n + #) &]];
a[ n_] := If[ n < 2, Boole[n >= 0], Times @@ (Which[ # == 2, 1, Mod[#, 4] == 1, #2 + 1, True, If[# == 3, 4, 1] Mod[#2 + 1, 2]] & @@@ FactorInteger[n])];
a[ n_] := SeriesCoefficient[ q^(1/8) QPochhammer[ -q^3] EllipticTheta[ 2, Pi/4, q^(3/2)]^3 / (Sqrt[2] EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(9/2)]), {q, 0, n}];
PROG
(PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};
(PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p%4==1, e+1, if( p==3, 4, 1) * (1 - e%2) )))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2 * eta(x^18 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^9 + A) * eta(x^36 + A)), n))};
(Magma) A := Basis( ModularForms( Gamma1(36), 1), 87); A[1] + A[2] + A[3] + A[5] + 2*A[6] + A[9] + 4*A[10] + 2*A[11] + 2*A[14] + A[17] + 2*A[18] + 4*A[19];
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Michael Somos, May 24 2015
STATUS
approved