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A145933 Expansion of chi(-q^6) * chi(-q^10) / (chi(-q) * chi(-q^15)) in powers of q where chi() is a Ramanujan theta function. 3
1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 11, 14, 17, 20, 23, 27, 31, 36, 41, 48, 55, 63, 72, 82, 94, 106, 122, 137, 156, 175, 197, 222, 249, 280, 314, 352, 393, 439, 490, 546, 608, 676, 751, 834, 923, 1024, 1133, 1253, 1384, 1528, 1686, 1857, 2045, 2250, 2474, 2718, 2984 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Michael Somos, Introduction to Ramanujan theta functions

Michael Somos, A Remarkable eta-product Identity

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of eta(q^2) * eta(q^6) * eta(q^10) * eta(q^30) / (eta(q) * eta(q^12) * eta(q^15) * eta(q^20)) in powers of q.

Euler transform of a period 60 sequence.

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 2 * (1 - u)^2 * (1 - v)^2 + 2 * u*v * (u - v)^2 + u*v * (v - u^2) * (u - v^2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A058728.

G.f.: Product_{k>0} (1 + x^k) * (1 + x^(15*k)) / ((1 + x^(6*k)) * (1 + x^(10*k))).

a(n) = A143752(n) unless n=0. a(2*n) = A094203(n).

Convolution inverse of A132968.

a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015

EXAMPLE

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

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ -x^15, x^15] / (QPochhammer[ -x^6, x^6] QPochhammer[ -x^10, x^10]), {x, 0, n}]; (* Michael Somos, Sep 03 2015 *)

nmax = 60; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(15*k)) / ((1 + x^(6*k)) * (1 + x^(10*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^6 + A) * eta(x^10 + A) * eta(x^30 + A) / (eta(x + A) * eta(x^12 + A) * eta(x^15 + A) * eta(x^20 + A)), n))};

CROSSREFS

Cf. A058728, A094203, A132968, A143752.

Sequence in context: A140952 A096911 A143752 * A300788 A120171 A145816

Adjacent sequences:  A145930 A145931 A145932 * A145934 A145935 A145936

KEYWORD

nonn

AUTHOR

Michael Somos, Oct 25 2008

STATUS

approved

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Last modified July 25 09:49 EDT 2021. Contains 346289 sequences. (Running on oeis4.)