A279476 Expansion of phi(-x^4) / (chi(-x^12) * f(-x)^2) in powers of x where phi(), chi(), f() are Ramanujan theta functions.
1, 2, 5, 10, 18, 32, 55, 90, 145, 228, 351, 532, 796, 1172, 1708, 2462, 3512, 4966, 6965, 9688, 13383, 18362, 25031, 33922, 45717, 61280, 81737, 108506, 143387, 188672, 247249, 322734, 419702, 543852, 702300, 903932, 1159779, 1483492, 1892012, 2406210, 3051796
Offset: 0
Keywords
Examples
G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 18*x^4 + 32*x^5 + 55*x^6 + ... G.f. = q^5 + 2*q^17 + 5*q^29 + 10*q^41 + 18*q^53 + 32*q^65 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..5000
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^4] QPochhammer[ -x^12, x^12] / QPochhammer[ x]^2 , {x, 0, n}]; a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^4] QPochhammer[ -x^12, x^12] / EllipticTheta[ 4, 0, x] , {x, 0, n}]; -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 * eta(x^24 + A) / (eta(x + A)^2 * eta(x^8 + A) * eta(x^12 + A)), n))}; -
PARI
q='q+O('q^99); Vec(eta(q^4)^2*eta(q^24)/(eta(q)^2*eta(q^8)*eta(q^12))) \\ Altug Alkan, Jul 30 2018
Formula
Expansion of chi(x^2) / (chi(-x^12) * phi(-x)) in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of q^(-5/12) * eta(q^4)^2 * eta(q^24) / (eta(q)^2 * eta(q^8) * eta(q^12)) in powers of q.
Euler transform of period 24 sequence [ 2, 2, 2, 0, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 0, 2, 2, 2, 1, ...].
G.f. is a period 1 Fourier series that satisfies f(-1 / (864 t)) = 288^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A279479.
a(n) = A131945(2*n + 1).
a(n) ~ sqrt(5) * exp(sqrt(10*n)*Pi/3) / (24*n). - Vaclav Kotesovec, Nov 29 2019
Comments