OFFSET
0,15
COMMENTS
Early in 2005 Michael Somos discovered a remarkable eta-product identity: eta(q^2) * eta(q^6) * eta(q^10) * eta(q^30) = eta(q) * eta(q^12) * eta(q^15) * eta(q^20) + eta(q^3) * eta(q^4) * eta(q^5) * eta(q^60).
G.f. is a period 1 Fourier series that satisfies f(-1 / (60 t)) = 60 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A286129. - Michael Somos, Oct 31 2019
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, A Remarkable eta-product Identity
FORMULA
G.f.: x^2 * Prod_{k>0} (1 - x^k) * (1 - x^(12 * k)) * (1 - x^(15 * k)) * (1 - x^(20 * k)).
EXAMPLE
G.f. = x^2 - x^3 - x^4 + x^7 + x^9 - 2*x^14 + x^15 + x^16 - 2*x^17 + x^18 + ... - Michael Somos, Oct 31 2019
MATHEMATICA
eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[eta[q]* eta[q^12]*eta[q^15]*eta[q^20], {q, 0, 50}], q] (* G. C. Greubel, Jul 29 2018 *)
PROG
(PARI) q='q+O('q^50); A = eta(q)*eta(q^12)*eta(q^15)*eta(q^20); concat([0, 0], Vec(A)) \\ G. C. Greubel, Jul 29 2018
(PARI) ({a(n) = my(A); n -= 2; if ( n < 0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^12 + A) * (eta(x^15 + A) * eta(x^20 + A), n))}; /* Michael Somos, Oct 31 2019 */
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, May 03 2017
STATUS
approved