A000713 EULER transform of 3, 2, 2, 2, 2, 2, 2, 2, ... (Formerly M2731 N1096)

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

0,2

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

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

Cf. A000041, A000716.

Row sums of triangle A093010.

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

Sequence in context: A136376 A099845 A036635 * A261325 A261446 A328539

Adjacent sequences: A000710 A000711 A000712 * A000714 A000715 A000716

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

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

Status

approved