A109368 Number of partitions of n into parts relatively prime to 42.

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 5, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 27, 30, 34, 37, 40, 44, 49, 54, 60, 65, 71, 78, 85, 94, 103, 112, 122, 132, 144, 158, 172, 186, 201, 218, 237, 258, 279, 302, 326, 352, 381, 412, 445, 480, 516, 556, 599, 646

Offset

0,6

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 = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Euler transform of period 42 sequence [ 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, ...].

G.f.: Product_{k>0} (1 + x^k) * (1 + x^(21*k)) / ((1 + x^(3*k)) * (1 + x^(7*k))) = Product_{k>0} P42(x^k) where P42 is the 42nd cyclotomic polynomial.

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

Expansion of q^(-1/2) * eta(q^2) * eta(q^3) * eta(q^7) * eta(q^42) / (eta(q) * eta(q^6) * eta(q^14) * eta(q^21)) in powers of q.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (168 t)) = 1 / f(t) where q = exp(2 Pi i t).

Example

1 + x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 2*x^8 + 2*x^9 + ...
q + q^3 + q^5 + q^7 + q^9 + 2*q^11 + 2*q^13 + 2*q^15 + 2*q^17 + 2*q^19 + ...
a(10) = 3 since 5 + 5 = 5 + 1 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 in 3 ways.

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/2)* eta[q^2]* eta[q^3]*eta[q^7]*eta[q^42]/(eta[q]*eta[q^6]*eta[q^14] *eta[q^21]), {q, 0, 50}], q] (* G. C. Greubel, Apr 18 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^7 + A) * eta(x^42 + A) / (eta(x + A) * eta(x^6 + A) * eta(x^14 + A) * eta(x^21 + A)), n))}

Crossrefs

Convolution inverse of A112211.

Sequence in context: A103373 A038539 A275891 * A046774 A029105 A079954

Adjacent sequences: A109365 A109366 A109367 * A109369 A109370 A109371

Keyword

nonn

Author

Michael Somos, Jun 26 2005, Jan 12 2009

Status

approved