A210705 Number of partitions of n into parts congruent to +-3, +-7, +-15, +-21, +-33, +-35, +-39, +-49, +-51, +-57 (mod 126).

1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 4, 2, 1, 4, 2, 1, 4, 4, 2, 5, 4, 2, 6, 4, 5, 8, 5, 5, 9, 6, 5, 13, 8, 6, 14, 9, 7, 15, 14, 10, 18, 15, 11, 21, 16, 17, 26, 19, 18, 29, 22, 19, 37, 28, 23, 41, 31, 27, 45, 41, 34, 53, 45, 38, 61

Offset

0,16

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 = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of chi(-x^9) * chi(-x^21) / (chi(-x^3) * chi(-x^7)) in powers of x where chi() is a Ramanujan theta function.

Expansion of (G(x^126) * H(x) - x^25 * G(x) * H(x^126)) / (G(x^63) * G(x^2) + x^13 * H(x^63) * H(x^2)) where G(x) and H(x) respectively are the g.f. of A003114 and A003106.

Expansion of q^(5/6) * eta(q^6) * eta(q^9) * eta(q^14) * eta(q^21) / ( eta(q^3) * eta(q^7) * eta(q^18) * eta(q^42) ) in powers of q.

Euler transform of a period 126 sequence.

G.f. is a period 1 Fourier series which satisfies f(-1 / (4536 t)) = f(t) where q = exp(2 Pi i t).

Example

1 + x^3 + x^6 + x^7 + x^9 + x^10 + x^12 + x^13 + x^14 + 2*x^15 + x^16 + ...
q^-5 + q^13 + q^31 + q^37 + q^49 + q^55 + q^67 + q^73 + q^79 + 2*q^85 + ...

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_]:= SeriesCoefficient[q^(5/6)* eta[q^6]*eta[q^9]*eta[q^14]*eta[q^21]/(eta[q^3]*eta[q^7]*eta[q^18] *eta[q^42]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 18 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A) * eta(x^9 + A) * eta(x^14 + A) * eta(x^21 + A) / ( eta(x^3 + A) * eta(x^7 + A) * eta(x^18 + A) * eta(x^42 + A) ), n))}

Crossrefs

Cf. A003106, A003114.

Sequence in context: A068696 A343388 A133009 * A291771 A186972 A373694

Adjacent sequences: A210702 A210703 A210704 * A210706 A210707 A210708

Keyword

nonn,changed

Author

Michael Somos, Aug 31 2013

Extensions

Name changed by David A. Corneth, Apr 18 2018

Status

approved