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A070048 Number of partitions of n into odd parts in which no part appears more than thrice.

%I A070048 #61 Jun 22 2025 07:12:11
%S A070048 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,
%T A070048 70,78,88,100,112,127,143,159,179,199,222,248,276,308,342,380,421,465,
%U A070048 516,570,629,697,767,845,932,1022,1124,1236,1355,1488,1631,1785,1954,2136
%N A070048 Number of partitions of n into odd parts in which no part appears more than thrice.
%C A070048 Also number of partitions of n into distinct parts in which no part is multiple of 4. - _Vladeta Jovovic_, Jul 31 2004
%C A070048 McKay-Thompson series of class 64a for the Monster group.
%C A070048 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A070048 Seiichi Manyama, <a href="/A070048/b070048.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..220 from Reinhard Zumkeller)
%H A070048 G. E. Andrews and R. P. Lewis, <a href="http://dx.doi.org/10.1016/S0012-365X(00)00295-8">An algebraic identity of F. H. Jackson and its implications for partitions</a>, Discrete Math., 232 (2001), 77-83.
%H A070048 Cristina Ballantine and Mircea Merca, <a href="https://arxiv.org/abs/2111.10702">4-Regular partitions and the pod function</a>, arXiv:2111.10702 [math.CO], 2021.
%H A070048 D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%H A070048 M. D. Hirschhorn, J. A. Sellers, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Sellers/sellers32.html">A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four</a>, Article 14.9.6, Journal of Integer Sequences, Vol. 17 (2014).
%H A070048 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 12.
%H A070048 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F A070048 G.f.: Product_{i>0} (1+x^i)/(1+x^(4*i)). - _Vladeta Jovovic_, Jul 31 2004
%F A070048 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
%F A070048 Expansion of q^(1/8) * eta(q^2) * eta(q^4) / (eta(q) * eta(q^8)) in powers of q.
%F A070048 Euler transform of period 8 sequence [1, 0, 1, -1, 1, 0, 1, 0, ...].
%F A070048 Given g.f. A(x), then B(q) = A(q^8) / q satisfies 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
%F A070048 G.f.: Product_{k>0} (1 - x^(8*k - 4)) / (1 - x^(2*k - 1)).
%F A070048 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
%e A070048 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 + ...
%e A070048 T64a = 1/q + q^7 + q^15 + 2*q^23 + q^31 + 2*q^39 + 3*q^47 + 3*q^55 + 4*q^63 + ...
%t A070048 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^2, x^4], {x, 0, n}]; (* _Michael Somos_, Jul 01 2014 *)
%t A070048 a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^4] / (QPochhammer[ x] QPochhammer[ x^8]), {x, 0, n}]; (* _Michael Somos_, Jul 01 2014 *)
%o A070048 (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))};
%o A070048 (Haskell)
%o A070048 a070048 = p a042968_list where
%o A070048    p _      0 = 1
%o A070048    p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
%o A070048 -- _Reinhard Zumkeller_, Oct 01 2012
%Y A070048 Cf. A042968, A001935, A261734.
%Y A070048 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).
%K A070048 nonn
%O A070048 0,4
%A A070048 _N. J. A. Sloane_, May 09 2002
%E A070048 Additional comments from _Michael Somos_, Dec 04 2002