A262950 Expansion of q * chi(-q^3) * chi(-q^13) / (chi(-q) * chi(-q^39)) in powers of q where chi() is a Ramanujan theta function.

1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 6, 6, 7, 8, 9, 11, 12, 14, 15, 17, 20, 22, 25, 28, 31, 34, 38, 43, 48, 53, 58, 64, 71, 79, 88, 96, 105, 115, 127, 140, 154, 168, 183, 200, 219, 240, 262, 285, 310, 337, 367, 400, 435, 472, 511, 554, 601, 652, 707, 764, 825

Offset

1,6

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 = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of eta(q^2) * eta(q^3) * eta(q^13) * eta(q^78) / (eta(q) * eta(q^6) * eta(q^26) * eta(q^39)) in powers of q.

Euler transform of a period 78 sequence.

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u^2 - v) * (v - w^2) - 2 * u*w * (1 + v)^2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (78 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128519.

G.f.: x * Product_{k>0} P(x^k) where P(x) is the 78th cyclotomic polynomial of degree 24.

Convolution inverse of A128519.

a(n) ~ exp(2*Pi*sqrt(2*n/39)) / (2^(3/4) * 39^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 29 2019

Example

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

Mathematica

eta[q_] := q^(1/24)*QPochhammer[q]; b := eta[q^2]*eta[q^3]*eta[q^13]* eta[q^78]/(eta[q]*eta[q^6]*eta[q^26]*eta[q^39]); a:= CoefficientList[ Series[q*b, {q, 0, 80}], q]; Drop[Table[a[[n]], {n, 1, 80}], 2] (* G. C. Greubel, Jul 03 2018 *)
nmax = 100; CoefficientList[Series[Product[(1 + x^k) * (1 + x^(39*k)) / ((1 + x^(3*k)) * (1 + x^(13*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 29 2019 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^13 + A) * eta(x^78 + A) / (eta(x + A) * eta(x^6 + A) * eta(x^26 + A) * eta(x^39 + A)), n))};

Crossrefs

Cf. A128519.

Sequence in context: A348336 A302984 A029094 * A227398 A003105 A240856

Adjacent sequences: A262947 A262948 A262949 * A262951 A262952 A262953

Keyword

nonn

Author

Michael Somos, Nov 19 2015

Status

approved