cp's OEIS Frontend

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

%I A000716 M2788 N1123 #120 Apr 24 2025 08:07:50
%S A000716 1,3,9,22,51,108,221,429,810,1479,2640,4599,7868,13209,21843,35581,
%T A000716 57222,90882,142769,221910,341649,521196,788460,1183221,1762462,
%U A000716 2606604,3829437,5590110,8111346,11701998,16790136,23964594,34034391,48104069,67679109,94800537,132230021,183686994,254170332
%N A000716 Number of partitions of n into parts of 3 kinds.
%D A000716 H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 122.
%D A000716 Moreno, Carlos J., Partitions, congruences and Kac-Moody Lie algebras. Preprint, 37pp., no date. See Table I.
%D A000716 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000716 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000716 Seiichi Manyama, <a href="/A000716/b000716.txt">Table of n, a(n) for n = 0..10000</a> (first 501 terms from T. D. Noe)
%H A000716 Roland Bacher and P. De La Harpe, <a href="https://hal.archives-ouvertes.fr/hal-01285685/document">Conjugacy growth series of some infinitely generated groups</a>, 2016, hal-01285685v2.
%H A000716 Victor J. W. Guo and Jiang Zeng, <a href="http://arxiv.org/abs/1205.4340">Two truncated identities of Gauss</a>, arXiv 1205.4340 [math.CO], 2012. - _N. J. A. Sloane_, Oct 10 2012
%H A000716 Masazumi Honda and Takuya Yoda, <a href="https://arxiv.org/abs/2203.17091">String theory, N = 4 SYM and Riemann hypothesis</a>, arXiv:2203.17091 [hep-th], 2022.
%H A000716 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=391">Encyclopedia of Combinatorial Structures 391</a>
%H A000716 Vladimir P. Kostov, <a href="http://arxiv.org/abs/1504.00883">Asymptotic expansions of zeros of a partial theta function</a>, arXiv:1504.00883 [math.CA], 2015.
%H A000716 Vladimir P. Kostov, <a href="http://arxiv.org/abs/1510.02584">Stabilization of the asymptotic expansions of the zeros of a partial theta function</a>, arXiv preprint arXiv:1510.02584 [math.CA], 2015.
%H A000716 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. 8.
%H A000716 P. Nataf, M. Lajkó, A. Wietek, K. Penc, F. Mila, and A. M. Läuchli, <a href="https://arxiv.org/abs/1601.00958">Chiral spin liquids in triangular lattice SU(N) fermionic Mott insulators with artificial gauge fields</a>, arXiv preprint arXiv:1601.00958 [cond-mat.quant-gas], 2016.
%H A000716 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A000716 <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>
%F A000716 G.f.: Product_{m>=1} 1/(1-x^m)^3.
%F A000716 EULER transform of 3, 3, 3, 3, 3, 3, 3, 3, ...
%F A000716 a(0)=1, a(n) = 1/n*Sum_{k=0..n-1} 3*a(k)*sigma_1(n-k). - _Joerg Arndt_, Feb 05 2011
%F A000716 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
%F A000716 G.f.: exp(3*Sum_{k>=1} x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 06 2018
%p A000716 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
%t A000716 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_ *)
%t A000716 (1/QPochhammer[q]^3 + O[q]^40)[[3]] (* _Vladimir Reshetnikov_, Nov 21 2016 *)
%o A000716 (PARI) Vec(1/eta('x+O('x^66))^3) \\ _Joerg Arndt_, Apr 28 2013
%o A000716 (Python)
%o A000716 from functools import lru_cache
%o A000716 from sympy import divisor_sigma
%o A000716 @lru_cache(maxsize=None)
%o A000716 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
%Y A000716 Cf. A000712.
%Y A000716 Column 3 of A144064.
%K A000716 nonn
%O A000716 0,2
%A A000716 _N. J. A. Sloane_
%E A000716 Extended with formula from _Christian G. Bower_, Apr 15 1998