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A061159 Numerators in expansion of Euler transform of b(n) = 1/2.

%I A061159 #33 Feb 03 2023 16:29:10
%S A061159 1,1,7,17,203,455,2723,6001,133107,312011,1613529,3705303,39159519,
%T A061159 88466147,443939867,1041952049,40842931395,93889422323,460998957853,
%U A061159 1054706036923,10194929714949,23513104814105,111438617932133,255719229005751,4864448363248503
%N A061159 Numerators in expansion of Euler transform of b(n) = 1/2.
%C A061159 Denominators of c(n) are 2^d(n), where d(n)=power of 2 in (2n)!, cf. A005187.
%H A061159 Alois P. Heinz, <a href="/A061159/b061159.txt">Table of n, a(n) for n = 0..1000</a>
%H A061159 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 A061159 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 A061159 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 A061159 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A061159 Numerators of c(n), where c(n)=1/(2*n)*Sum_{k=1..n} c(n-k)*sigma(k), n>0, c(0)=1.
%p A061159 b:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A061159       d/2, d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
%p A061159     end:
%p A061159 a:= n-> numer(b(n)):
%p A061159 seq(a(n), n=0..35);  # _Alois P. Heinz_, Jul 28 2017
%t A061159 c[n_] := c[n] = If[n == 0, 1,
%t A061159      (1/(2n)) Sum[c[n-k] DivisorSigma[1, k], {k, 1, n}]];
%t A061159 a[n_] := Numerator[c[n]];
%t A061159 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, Apr 24 2022 *)
%Y A061159 Cf. A000712, A061160, A061161.
%K A061159 easy,nonn,frac
%O A061159 0,3
%A A061159 _Vladeta Jovovic_, Apr 17 2001