OFFSET
1,2
COMMENTS
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
Xuhang Jiang, Xing Wang, Li Lin Yang and Jing-Bang Zhao, Epsilon-factorized differential equations for two-loop non-planar triangle Feynman integrals with elliptic curves, arXiv:2305.13951 [hep-th], 2023. See page 21.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q / (chi(-q^3)^3 / chi(-q))^3 in powers of q where chi() is a Ramanujan theta function.
Euler transform of period 6 sequence [ -3, 0, 6, 0, -3, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v)= u^2 - v - u*v * (6 + 8*v).
G.f.: x * (Product_{k>0} (1 - x^(2*k - 1)) / (1 - x^(6*k - 3))^3 )^3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (1 / 8) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128642.
Convolution cube of A092848.
EXAMPLE
G.f. = q - 3*q^2 + 3*q^3 + 5*q^4 - 18*q^5 + 15*q^6 + 24*q^7 - 75*q^8 + 57*q^9 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ q / (QPochhammer[ q^3, q^6]^3 / QPochhammer[ q, q^2])^3, {q, 0, n}]; (* Michael Somos, Feb 19 2015 *)
a[ n_] := SeriesCoefficient[ q (Product[ 1 - q^k, {k, 1, n, 2}] / Product[ 1 - q^k, {k, 3, n, 6}]^3)^3, {q, 0, n}]; (* Michael Somos, Feb 19 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^3 * (eta(x^6 + A) / eta(x^3 + A))^9, n))};
(Magma) M := Basis(ModularForms(Gamma1(6), 1), 43); M1 := M[1]; M2 := M[2]; A<q> := M2/(M1 + 2*M2); A; /* Michael Somos, May 26 2023 */
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 03 2006, Jan 21 2009
STATUS
approved