A146162 -id:A146162 - cp's OEIS frontend

A146164 Expansion of f(-x^4) * chi(x^5) / f(-x^5) in powers of x where f(), chi() are Ramanujan theta functions.

1, 0, 0, 0, -1, 2, 0, 0, -1, -2, 3, 0, 0, -2, -3, 6, 0, 0, -3, -6, 11, 0, 0, -6, -10, 18, 0, 0, -9, -16, 28, 0, 0, -14, -25, 44, 0, 0, -22, -38, 67, 0, 0, -32, -57, 100, 0, 0, -48, -84, 146, 0, 0, -70, -121, 210, 0, 0, -99, -172, 299, 0, 0, -140, -243, 420, 0

Offset: 0

Michael Somos, Oct 27 2008, Nov 10 2008

sign

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

			G.f. = 1 - x^4 + 2*x^5 - x^8 - 2*x^9 + 3*x^10 - 2*x^13 - 3*x^14 + 6*x^15 + ...
G.f. = 1/q - q^15 + 2*q^19 - q^31 - 2*q^35 + 3*q^39 - 2*q^51 - 3*q^55 + ...

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. A138532, A146162, A146163, A146165.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^4] QPochhammer[ -x^5, x^10] / QPochhammer[ x^5], {x, 0, n}]; (* Michael Somos, Sep 03 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^10 + A)^2 / (eta(x^5 + A)^2 * eta(x^20 + A)), n))};

Expansion of q^(1/4) * eta(q^4) * eta(q^10)^2 / (eta(q^5)^2 * eta(q^20)) in powers of q.
Euler transform of period 20 sequence [ 0, 0, 0, -1, 2, 0, 0, -1, 0, 0, 0, -1, 0, 0, 2, -1, 0, 0, 0, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = (5/4)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A146162.
a(5*n + 1) = a(5*n + 2) = 0.
a(n) = A138532(2*n + 1). a(5*n + 4) = - A146163(n).
Convolution inverse of A146165.

Edited by N. J. A. Sloane, Nov 21 2008 at the suggestion of R. J. Mathar

A146163 Expansion of q^(-3/4) * eta(q^2)^2 * eta(q^20) / (eta(q)^2 * eta(q^4)) in powers of q.

Original entry on oeis.org

1, 2, 3, 6, 10, 16, 25, 38, 57, 84, 121, 172, 243, 338, 465, 636, 862, 1158, 1546, 2050, 2701, 3540, 4613, 5980, 7719, 9916, 12682, 16158, 20506, 25926, 32667, 41022, 51348, 64080, 79730, 98922, 122407, 151068, 185968, 228384, 279816, 342052

Offset: 0

Michael Somos, Oct 27 2008

nonn

			q^3 + 2*q^7 + 3*q^11 + 6*q^15 + 10*q^19 + 16*q^23 + 25*q^27 + 38*q^31 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000

nmax = 50; CoefficientList[Series[Product[(1+x^k)^2 * (1-x^(20*k)) / (1-x^(4*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 11 2016 *)
a[n_]:= SeriesCoefficient[QPochhammer[-q, q]^2*QPochhammer[q^20, q^20]/(QPochhammer[q^4, q^4]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Dec 05 2017 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^20 + A) / (eta(x + A)^2 * eta(x^4 + A)), n))}

Euler transform of period 20 sequence [ 2, 0, 2, 1, 2, 0, 2, 1, 2, 0, 2, 1, 2, 0, 2, 1, 2, 0, 2, 0, ...].
a(n) ~ exp(2*Pi*sqrt(n/5)) / (4*5^(3/4)*n^(3/4)). - Vaclav Kotesovec, Jul 11 2016
a(n) = A146162(4*n + 3).

cp's OEIS Frontend

A146164 Expansion of f(-x^4) * chi(x^5) / f(-x^5) in powers of x where f(), chi() are Ramanujan theta functions.

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