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%I M2731 N1096 #60 Sep 25 2023 14:26:38
%S 1,3,8,18,38,74,139,249,434,734,1215,1967,3132,4902,7567,11523,17345,
%T 25815,38045,55535,80377,115379,164389,232539,326774,456286,633373,
%U 874213,1200228,1639418,2228546,3015360,4062065,5448995,7280060,9688718,12846507,16972577
%N EULER transform of 3, 2, 2, 2, 2, 2, 2, 2, ...
%C Equals row sums of triangle A146023. - _Gary W. Adamson_, Oct 26 2008
%C Partial sums of A000712. - _Geoffrey Critzer_, Apr 19 2012, corrected by _Omar E. Pol_, Jun 19 2012
%C 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
%D H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 122.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000713/b000713.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=390">Encyclopedia of Combinatorial Structures 390</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F G.f.: A(x)/(1-x) where A(x) is g.f. for A000712. - _Geoffrey Critzer_, Apr 19 2012.
%F From _Vaclav Kotesovec_, Aug 16 2015: (Start)
%F a(n) ~ sqrt(3*n)/Pi * A000712(n).
%F a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*Pi*3^(1/4)*n^(3/4)).
%F (End)
%F G.f.: exp(Sum_{k>=1} (2*sigma_1(k) + 1)*x^k/k). - _Ilya Gutkovskiy_, Aug 21 2018
%p 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
%t 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 *)
%o (PARI) x='x+O('x^66); Vec(1/((1-x)*eta(x)^2)) \\ _Joerg Arndt_, May 01 2013
%o (Python)
%o from functools import lru_cache
%o from sympy import divisor_sigma
%o @lru_cache(maxsize=None)
%o 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
%Y Cf. A000041, A000716.
%Y Row sums of triangle A093010.
%Y Cf. A146023. - _Gary W. Adamson_, Oct 26 2008
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E Extended with formula from _Christian G. Bower_, Apr 15 1998
%E Definition changed by _N. J. A. Sloane_, Aug 15 2006
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