OFFSET
0,2
COMMENTS
A000712: (1, 2, 5, 10, 20, 36, ...) = this sequence convolved with A010815. - Gary W. Adamson, Oct 26 2008
It appears that the partial sums give A210843. - Omar E. Pol, Jun 18 2012
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
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Extended with formula from Christian G. Bower, Apr 15 1998
STATUS
approved