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 A070048 Number of partitions of n into odd parts in which no part appears more than thrice. 16
 1, 1, 1, 2, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 11, 13, 16, 18, 21, 24, 27, 32, 36, 41, 48, 54, 61, 70, 78, 88, 100, 112, 127, 143, 159, 179, 199, 222, 248, 276, 308, 342, 380, 421, 465, 516, 570, 629, 697, 767, 845, 932, 1022, 1124, 1236, 1355, 1488, 1631, 1785, 1954, 2136 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Also number of partitions of n into distinct parts in which no part is multiple of 4. - Vladeta Jovovic, Jul 31 2004 McKay-Thompson series of class 64a for the Monster group. Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..220 from Reinhard Zumkeller) G. E. Andrews and R. P. Lewis, An algebraic identity of F. H. Jackson and its implications for partitions, Discrete Math., 232 (2001), 77-83. D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994). M. D. Hirschhorn, J. A. Sellers, A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four, Article 14.9.6, Journal of Integer Sequences, Vol. 17 (2014). Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 12. FORMULA G.f.: Product_{i>0} (1+x^i)/(1+x^(4*i)). - Vladeta Jovovic, Jul 31 2004 Expansion of chi(x) * chi(x^2) = psi(x) / psi(-x^2) = phi(-x^4) / psi(-x) = chi(-x^4) / chi(-x) in powers of x where phi(), psi(), chi() are Ramanujan theta functions. - Michael Somos, Jul 01 2014 Expansion of q^(1/8) * eta(q^2) * eta(q^4) / (eta(q) * eta(q^8)) in powers of q. Euler transform of period 8 sequence [1, 0, 1, -1, 1, 0, 1, 0, ...]. Given g.f. A(x), then B(q) = A(q^8) / q satisifes 0 = f(B(q), B(q^3)) where f(u, v) = (u - v^3) * (u^3 - v) + 3*u*v. - Michael Somos, Jul 01 2014 G.f.: Product_{k>0} (1 - x^(8*k - 4)) / (1 - x^(2*k - 1)). a(n) ~ exp(sqrt(n)*Pi/2) / (4*n^(3/4)) * (1 - (3/(4*Pi) + Pi/32) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017 EXAMPLE G.f. = 1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 5*x^9 + ... T64a = 1/q + q^7 + q^15 + 2*q^23 + q^31 + 2*q^39 + 3*q^47 + 3*q^55 + 4*q^63 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^2, x^4]), {x, 0, n}]; (* Michael Somos, Jul 01 2014 *) a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^4] / (QPochhammer[ x] QPochhammer[ x^8]), {x, 0, n}]; (* Michael Somos, Jul 01 2014 *) PROG (PARI) {a(n) = local(A); if( n<0, 0 , A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A) / (eta(x + A) * eta(x^8 + A)), n))}; (Haskell) a070048 = p a042968_list where    p _      0 = 1    p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m -- Reinhard Zumkeller, Oct 01 2012 CROSSREFS Cf. A042968, A001935, A261734. Cf. A000700 (m=2), A003105 (m=3), A096938 (m=5), A261770 (m=6), A097793 (m=7), A261771 (m=8), A112193 (m=9), A261772 (m=10). Sequence in context: A239511 A264052 A138585 * A116498 A143472 A180235 Adjacent sequences:  A070045 A070046 A070047 * A070049 A070050 A070051 KEYWORD nonn AUTHOR N. J. A. Sloane, May 09 2002 EXTENSIONS Additional comments from Michael Somos, Dec 04 2002 STATUS approved

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Last modified October 19 16:17 EDT 2019. Contains 328223 sequences. (Running on oeis4.)