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A000713 EULER transform of 3, 2, 2, 2, 2, 2, 2, 2, ...
(Formerly M2731 N1096)
7

%I M2731 N1096

%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

%Y Cf. 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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Last modified July 15 04:00 EDT 2020. Contains 335762 sequences. (Running on oeis4.)