A224822 -id:A224822 - cp's OEIS frontend

A329955 Expansion of eta(q) * eta(q^2) * eta(q^3)^3 / eta(q^6)^2 in powers of q.

1, -1, -2, -2, 3, 8, 0, -2, -10, -4, 2, 4, 10, -8, -4, 0, 7, 12, 4, -2, -16, -16, 4, 8, 0, -7, -4, -2, 10, 24, 8, -2, -26, 0, 2, 8, 12, -16, -8, -8, 10, 12, 0, -6, -20, -16, 4, 8, 26, -7, -10, 0, 16, 40, 0, -4, -20, -24, 6, 4, 0, -16, -12, -8, 15, 24, 8, -6

Offset: 0

Michael Somos, Nov 26 2019

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

Cf. A028967, A030206, A195848, A224822, A329956, A329957.

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^2] QPochhammer[ x^3]^3 / QPochhammer[ x^6]^2, {x, 0, n}];

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

Euler transform of period 6 sequence [-1, -2, -4, -2, -1, -3, ...].
G.f.: Product_{k>=1} (1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) / (1 + x^(3*k))^2.
Convolution of A030206 and A195848.
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 1990656^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A329958.
a(3*n) = A224822(n). a(3*n + 1) = -A329956(n). a(3*n + 2) = -2*A329957(n). a(6*n) = A028967(n).

A224833 Expansion of phi(-x)^2 * chi(-x) * psi(x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, -5, 8, -4, 4, -13, 12, -4, 5, -16, 24, -8, 4, -20, 12, -8, 9, -20, 32, -4, 12, -29, 12, -8, 8, -36, 40, -8, 8, -20, 24, -16, 8, -25, 40, -12, 12, -32, 24, -12, 13, -48, 40, -8, 8, -40, 36, -8, 16, -20, 56, -16, 12, -52, 12, -20, 13, -36, 56, -16, 20, -40, 24

Offset: 0

Michael Somos, Jul 21 2013

sign

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

			1 - 5*x + 8*x^2 - 4*x^3 + 4*x^4 - 13*x^5 + 12*x^6 - 4*x^7 + 5*x^8 - 16*x^9 + ...
q - 5*q^4 + 8*q^7 - 4*q^10 + 4*q^13 - 13*q^16 + 12*q^19 - 4*q^22 + 5*q^25 + ...

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. A224822, A227595.

eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(-1/3)* eta[q]^5*eta[q^6]^2/(eta[q^2]^3*eta[q^3]), {q, 0, n}];  Table[a[n], {n,0,50}] (* G. C. Greubel, Mar 19 2018 *)

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

Expansion of q^(-1/3) * eta(q)^5 * eta(q^6)^2 / (eta(q^2)^3 * eta(q^3)) in powers of q.
Euler transform of period 6 sequence [ -5, -2, -4, -2, -5, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 73728^(1/2) (t / i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227595.
-2 * a(n) = A224822(3*n + 1).

cp's OEIS Frontend

A329955 Expansion of eta(q) * eta(q^2) * eta(q^3)^3 / eta(q^6)^2 in powers of q.

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