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

1, 1, 7, 17, 203, 455, 2723, 6001, 133107, 312011, 1613529, 3705303, 39159519, 88466147, 443939867, 1041952049, 40842931395, 93889422323, 460998957853, 1054706036923, 10194929714949, 23513104814105, 111438617932133, 255719229005751, 4864448363248503

Offset: 0

Author

Vladeta Jovovic, Apr 17 2001

Keywords

• easy
• nonn
• frac

Comments

Denominators of c(n) are 2^d(n), where d(n)=power of 2 in (2n)!, cf. A005187.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Geoffrey B. Campbell, Some n-space q-binomial theorem extensions and similar identities, arXiv:1906.07526 [math.NT], 2019.
• Geoffrey B. Campbell, Continued Fractions for partition generating functions, arXiv:2301.12945 [math.CO], 2023.
• Geoffrey B. Campbell and A. Zujev, Some almost partition theoretic identities, Preprint, 2016.
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000712, A061160, A061161.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(add(
      d/2, d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
    end:
a:= n-> numer(b(n)):
seq(a(n), n=0..35);  # Alois P. Heinz, Jul 28 2017

Mathematica

c[n_] := c[n] = If[n == 0, 1,
     (1/(2n)) Sum[c[n-k] DivisorSigma[1, k], {k, 1, n}]];
a[n_] := Numerator[c[n]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Apr 24 2022 *)

Formula

Numerators of c(n), where c(n)=1/(2*n)*Sum_{k=1..n} c(n-k)*sigma(k), n>0, c(0)=1.