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

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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

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

Example

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 + ...

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}];

Prog

(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

Crossrefs

Cf. A131945, A279479.

Sequence in context: A079006 A001936 A246712 * A281683 A224364 A327064

Adjacent sequences: A279473 A279474 A279475 * A279477 A279478 A279479

Keyword

nonn

Author

Michael Somos, Dec 12 2016

Status

approved