A000716 Number of partitions of n into parts of 3 kinds.

1, 3, 9, 22, 51, 108, 221, 429, 810, 1479, 2640, 4599, 7868, 13209, 21843, 35581, 57222, 90882, 142769, 221910, 341649, 521196, 788460, 1183221, 1762462, 2606604, 3829437, 5590110, 8111346, 11701998, 16790136, 23964594, 34034391, 48104069, 67679109, 94800537, 132230021, 183686994, 254170332

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

• H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 122.
• Moreno, Carlos J., Partitions, congruences and Kac-Moody Lie algebras. Preprint, 37pp., no date. See Table I.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 501 terms from T. D. Noe)
• Roland Bacher and P. De La Harpe, Conjugacy growth series of some infinitely generated groups, 2016, hal-01285685v2.
• Victor J. W. Guo and Jiang Zeng, Two truncated identities of Gauss, arXiv 1205.4340 [math.CO], 2012. - _N. J. A. Sloane_, Oct 10 2012
• Masazumi Honda and Takuya Yoda, String theory, N = 4 SYM and Riemann hypothesis, arXiv:2203.17091 [hep-th], 2022.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 391
• Vladimir P. Kostov, Asymptotic expansions of zeros of a partial theta function, arXiv:1504.00883 [math.CA], 2015.
• Vladimir P. Kostov, Stabilization of the asymptotic expansions of the zeros of a partial theta function, arXiv preprint arXiv:1510.02584 [math.CA], 2015.
• 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. 8.
• P. Nataf, M. Lajkó, A. Wietek, K. Penc, F. Mila, and A. M. Läuchli, Chiral spin liquids in triangular lattice SU(N) fermionic Mott insulators with artificial gauge fields, arXiv preprint arXiv:1601.00958 [cond-mat.quant-gas], 2016.
• N. J. A. Sloane, Transforms
• Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Crossrefs

Cf. A000712.

Column 3 of A144064.

Programs

Maple

with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*3, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, May 20 2013

Mathematica

a[0] = 1; a[n_] := a[n] = 1/n*Sum[3*a[k]*DivisorSigma[1, n-k], {k, 0, n-1}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 03 2014, after Joerg Arndt *)
(1/QPochhammer[q]^3 + O[q]^40)[[3]] (* Vladimir Reshetnikov, Nov 21 2016 *)

PARI

Vec(1/eta('x+O('x^66))^3) \\ Joerg Arndt, Apr 28 2013

Python

from functools import lru_cache
from sympy import divisor_sigma
@lru_cache(maxsize=None)
def A000716(n): return sum(A000716(k)*divisor_sigma(n-k) for k in range(n))*3//n if n else 1 # Chai Wah Wu, Sep 25 2023

Formula

G.f.: Product_{m>=1} 1/(1-x^m)^3.

EULER transform of 3, 3, 3, 3, 3, 3, 3, 3, ...

a(0)=1, a(n) = 1/n*Sum_{k=0..n-1} 3*a(k)*sigma_1(n-k). - Joerg Arndt, Feb 05 2011

a(n) ~ exp(Pi * sqrt(2*n)) / (8 * sqrt(2) * n^(3/2)) * (1 - (3/Pi + Pi/8) / sqrt(2*n)). - Vaclav Kotesovec, Feb 28 2015, extended Jan 16 2017

G.f.: exp(3*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

Extensions

Extended with formula from Christian G. Bower, Apr 15 1998