OFFSET
0,2
COMMENTS
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
Expansion of eta(q^2)^5 * eta(q^18)^5 / (eta(q) * eta(q^4) * eta(q^9) * eta(q^36))^2 in powers of q.
Euler transform of period 36 sequence [2, -3, 2, -1, 2, -3, 2, -1, 4, -3, 2, -1, 2, -3, 2, -1, 2, -6, 2, -1, 2, -3, 2, -1, 2, -3, 4, -1, 2, -3, 2, -1, 2, -3, 2, -2, ...].
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(n) = (-1)^n * A258322(n). a(4*n) = a(n).
a(3*n + 2) = a(4*n + 3) = a(8*n + 6) = a(9*n + 3) = a(9*n + 6) = 0.
a(3*n + 1) = 2 * A122865(n). a(6*n + 4) = 2 * A122856(n). a(9*n) = A004018(n). a(12*n + 1) = 2 * A002175(n).
a(2*n) = A028601(n). - Michael Somos, Jul 04 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 (A019670). - Amiram Eldar, Jan 29 2024
EXAMPLE
G.f. = 1 + 2*q + 2*q^4 + 4*q^9 + 4*q^10 + 4*q^13 + 2*q^16 + 4*q^18 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^9], {q, 0, n}];
a[ n_] := Which[ n < 1, Boole[n == 0], Mod[n, 3] == 2, 0, True, 2 DivisorSum[ n, If[ Mod[n/#, 9] > 0, 1, 2] KroneckerSymbol[ -4, #] &]]; (* Michael Somos, Jul 04 2015 *)
PROG
(PARI) {a(n) = if( n<1, n==0, (n+1)%3 * 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); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^18 + A)^5 / (eta(x + A) * eta(x^4 + A) * eta(x^9 + A) * eta(x^36 + A))^2, n))};
(PARI) {a(n) = if( n<1, n==0, n%3==2, 0, 2 * sumdiv(n, d, if(n\d%9, 1, 2) * kronecker( -4, d)))}; /* Michael Somos, Jul 04 2015 */
(PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); (n%3 < 2) * 2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, 1 + (-1)^e, p%12>6, (1 + (-1)^e) / 2, e+1)))}; /* Michael Somos, Jul 04 2015 */
(Magma) A := Basis( ModularForms( Gamma1(36), 1), 87); A[1] + 2*A[2] + 2*A[5] + 4*A[10] + 4*A[11] + 4*A[14] + 2*A[17] + 4*A[19];
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jun 03 2015
STATUS
approved