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

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

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.

Cristina Ballantine and Mircea Merca, 4-Regular partitions and the pod function, arXiv:2111.10702 [math.CO], 2021.

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.

Index entries for McKay-Thompson series for Monster simple group

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