A254346 -id:A254346 - cp's OEIS frontend

A254372 Expansion of phi(q) * phi(-q^3) * f(-q^12) / f(-q^4)^3 in powers of q where phi(), f() are Ramanujan theta functions.

1, 2, 0, -2, 1, 6, 0, -10, 3, 20, 0, -30, 1, 52, 0, -78, 6, 126, 0, -184, 3, 280, 0, -402, 12, 590, 0, -830, 5, 1182, 0, -1636, 21, 2280, 0, -3108, 10, 4252, 0, -5722, 36, 7710, 0, -10252, 15, 13632, 0, -17940, 60, 23586, 0, -30744, 26, 40014, 0, -51714, 96

Offset: 0

Michael Somos, Jan 29 2015

sign

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

			G.f. = 1 + 2*q - 2*q^3 + q^4 + 6*q^5 - 10*q^7 + 3*q^8 + 20*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A132180, A254346, A230256.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] * EllipticTheta[ 4, 0, q^3] QPochhammer[ q^12] / QPochhammer[ q^4]^3, {q, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^2 * eta(x^12 + A) / (eta(x + A)^2 * eta(x^4 + A)^5 * eta(x^6 + A)), n))};

Expansion of eta(q^2)^5 * eta(q^3)^2 * eta(q^12) / (eta(q)^2 * eta(q^4)^5 * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 2, -3, 0, 2, 2, -4, 2, 2, 0, -3, 2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = (4/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A230256.
a(4*n + 2) = 0. a(2*n + 1) = 2 * A254346(n). a(4*n) = A132180(n).

A254525 Expansion of f(-x^2)^2 * f(-x, x^2) / f(x^3)^3 in powers of x where f(,) is Ramanujan's general theta function.

Original entry on oeis.org

1, -1, -1, -1, 0, 3, 4, -1, -6, -5, 1, 10, 11, -4, -19, -17, 4, 31, 31, -9, -50, -46, 11, 79, 77, -21, -122, -112, 28, 183, 173, -46, -273, -249, 62, 396, 370, -98, -573, -521, 130, 815, 751, -193, -1149, -1041, 261, 1599, 1461, -373, -2214, -1998, 498, 3031

Offset: 0

Michael Somos, Jan 31 2015

sign

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

			G.f. = 1 - x - x^2 - x^3 + 3*x^5 + 4*x^6 - x^7 - 6*x^8 - 5*x^9 + x^10 + ...
G.f. = 1/q - q^5 - q^11 - q^17 + 3*q^29 + 4*q^35 - q^41 - 6*q^47 - 5*q^53 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A062243, A128111, A132179, A230256, A233037, A254346.

eta[q_] := q^(1/24)*QPochhammer[q]; A254525[n_] := SeriesCoefficient[
   q^(1/6)*eta[q]*eta[q^3]*eta[q^4]*eta[q^12]/eta[q^6]^4, {q, 0, n}]; Table[A254525[n], {n,0,50}] (* G. C. Greubel, Aug 10 2017 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / eta(x^6 + A)^4, n))};

Expansion of (chi(x) / chi(x^3)^3) * (psi(-x) / psi(-x^3))^2 in powers of x where chi(), psi() are Ramanujan theta functions.
Expansion of q^(1/6) * eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / eta(q^6)^4 in powers of q.
Euler transform of period 12 sequence [ -1, -1, -2, -2, -1, 2, -1, -2, -2, -1, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 9^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A254346.
Convolution of A062243 and A128111.
a(n) = (-1)^n * A132179(n).
a(2*n) = A230256(n). a(2*n + 1) = - A233037(n).

cp's OEIS Frontend

A254372 Expansion of phi(q) * phi(-q^3) * f(-q^12) / f(-q^4)^3 in powers of q where phi(), f() are Ramanujan theta functions.

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