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 A210705 Number of partitions of n into parts congruent to +-3, +-7, +-15, +-21, +-33, +-35, +-39, +-49, +-51, +-57 (mod 126). 1
 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 (list; graph; refs; listen; history; text; internal format)
 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 A053734 Adjacent sequences: A210702 A210703 A210704 * A210706 A210707 A210708 KEYWORD nonn AUTHOR Michael Somos, Aug 31 2013 EXTENSIONS Name changed by David A. Corneth, Apr 18 2018 STATUS approved

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Last modified July 15 10:49 EDT 2024. Contains 374332 sequences. (Running on oeis4.)