A061161 Numerators in expansion of Euler transform of b(n) = 1/4.

1, 1, 13, 55, 1235, 4615, 55801, 200343, 8977475, 36804235, 367235363, 1444888289, 32062742231, 120729974115, 1205864254225, 5201022002071, 395884671433315, 1603069490974835, 15989295873680415, 64312573140322525, 1250332447587844829, 5262481040435242585

Offset: 0

Author

Vladeta Jovovic, Apr 17 2001

Keywords

• easy
• nonn
• frac

Comments

Denominators of c(n) are 2^A004134(n).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Geoffrey B. Campbell and A. Zujev, Some almost partition theoretic identities, Preprint, 2016.
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000712, A061159, A061160.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(add(
      d/4, d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
    end:
a:= n-> numer(b(n)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 28 2017

Mathematica

c[n_] := c[n] = If[n == 0, 1,
     (1/(4n)) Sum[c[n-k] DivisorSigma[1, k], {k, 1, n}]];
a[n_] := Numerator[c[n]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Apr 24 2022 *)

Formula

Numerators of c(n), where c(n) = (1/(4*n))*Sum_{k=1..n} c(n-k)*sigma(k), n>0, c(0)=1.