A000713 EULER transform of 3, 2, 2, 2, 2, 2, 2, 2, ...

1, 3, 8, 18, 38, 74, 139, 249, 434, 734, 1215, 1967, 3132, 4902, 7567, 11523, 17345, 25815, 38045, 55535, 80377, 115379, 164389, 232539, 326774, 456286, 633373, 874213, 1200228, 1639418, 2228546, 3015360, 4062065, 5448995, 7280060, 9688718, 12846507, 16972577

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Equals row sums of triangle A146023. - Gary W. Adamson, Oct 26 2008

Partial sums of A000712. - Geoffrey Critzer, Apr 19 2012, corrected by Omar E. Pol, Jun 19 2012

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

Crossrefs

Cf. A000041, A000716.

Row sums of triangle A093010.

Cf. A146023. - Gary W. Adamson, Oct 26 2008

Programs

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 *)

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

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

Extensions

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

Definition changed by N. J. A. Sloane, Aug 15 2006