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A061160 Numerators in expansion of Euler transform of b(n) = 1/3.

%I A061160 #38 Feb 07 2023 08:38:43
%S A061160 1,1,5,50,215,646,8711,25475,105925,3091270,11691247,36809705,
%T A061160 445872155,1364113925,5085042010,50975292560,183383680088,
%U A061160 588817265695,19512559194875,62369303509475,224877933068647,2214198452392027,7686538660149565,25124342108522750
%N A061160 Numerators in expansion of Euler transform of b(n) = 1/3.
%C A061160 Denominators of c(n) are 3^d(n), where d(n)=power of 3 in (3*n)!, cf. A004128.
%H A061160 Alois P. Heinz, <a href="/A061160/b061160.txt">Table of n, a(n) for n = 0..1000</a>
%H A061160 Geoffrey B. Campbell, <a href="https://arxiv.org/abs/1906.07526">Some n-space q-binomial theorem extensions and similar identities</a>, arXiv:1906.07526 [math.NT], 2019.
%H A061160 Geoffrey B. Campbell, <a href="https://arxiv.org/abs/2301.12945">Continued Fractions for partition generating functions</a>, arXiv:2301.12945 [math.CO], 2023.
%H A061160 Geoffrey B. Campbell, <a href="https://arxiv.org/abs/2302.01091">Vector Partition Identities for 2D, 3D and nD Lattices</a>, arXiv:2302.01091 [math.CO], 2023.
%H A061160 Geoffrey B. Campbell and A. Zujev, <a href="http://zujev.physics.ucdavis.edu/papers/Some%20almost%20partition%20theoretic%20identities.pdf">Some almost partition theoretic identities</a>, Preprint, 2016.
%H A061160 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A061160 Numerators of c(n), where c(n) = (1/(3*n))*Sum_{k=1..n} c(n-k)*sigma(k), n>0, c(0)=1.
%p A061160 b:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A061160       d/3, d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
%p A061160     end:
%p A061160 a:= n-> numer(b(n)):
%p A061160 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jul 28 2017
%t A061160 c[n_] := c[n] = If[n == 0, 1,
%t A061160      (1/(3n)) Sum[c[n-k] DivisorSigma[1, k], {k, 1, n}]];
%t A061160 a[n_] := Numerator[c[n]];
%t A061160 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Apr 24 2022 *)
%Y A061160 Cf. A000712, A061159, A061161.
%K A061160 easy,nonn,frac
%O A061160 0,3
%A A061160 _Vladeta Jovovic_, Apr 17 2001