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A109368 Number of partitions of n into parts relatively prime to 42. 2
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 (list; graph; refs; listen; history; text; internal format)
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

M. 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

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Last modified November 13 02:59 EST 2019. Contains 329085 sequences. (Running on oeis4.)