OFFSET
0,2
COMMENTS
Equals row sums of triangle A146023. - Gary W. Adamson, Oct 26 2008
Equals the number of partitions of n with 1's of three kinds and all parts >1 of two kinds. - Gregory L. Simay, Mar 25 2018
REFERENCES
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 122.
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
T. D. Noe, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 390
N. J. A. Sloane, Transforms
FORMULA
G.f.: A(x)/(1-x) where A(x) is g.f. for A000712. - Geoffrey Critzer, Apr 19 2012.
From Vaclav Kotesovec, Aug 16 2015: (Start)
a(n) ~ sqrt(3*n)/Pi * A000712(n).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*Pi*3^(1/4)*n^(3/4)).
(End)
G.f.: exp(Sum_{k>=1} (2*sigma_1(k) + 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018
MAPLE
with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> `if`(n<2, 3, 2)): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008
MATHEMATICA
nn=20; g=Product[1/(1-x^i), {i, 1, nn}]; c=1/(1-x); CoefficientList[Series[g^2/(1-x), {x, 0, nn}], x] (* Geoffrey Critzer, Apr 19 2012 *)
PROG
(PARI) x='x+O('x^66); Vec(1/((1-x)*eta(x)^2)) \\ Joerg Arndt, May 01 2013
(Python)
from functools import lru_cache
from sympy import divisor_sigma
@lru_cache(maxsize=None)
def A000713(n): return sum(A000713(k)*((divisor_sigma(n-k)<<1)+1) for k in range(n))//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
Definition changed by N. J. A. Sloane, Aug 15 2006
STATUS
approved