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