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

1, 1, 5, 50, 215, 646, 8711, 25475, 105925, 3091270, 11691247, 36809705, 445872155, 1364113925, 5085042010, 50975292560, 183383680088, 588817265695, 19512559194875, 62369303509475, 224877933068647, 2214198452392027, 7686538660149565, 25124342108522750

Offset: 0

Author

Vladeta Jovovic, Apr 17 2001

Keywords

• easy
• nonn
• frac

Comments

Denominators of c(n) are 3^d(n), where d(n)=power of 3 in (3*n)!, cf. A004128.

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, Vector Partition Identities for 2D, 3D and nD Lattices, arXiv:2302.01091 [math.CO], 2023.
• Geoffrey B. Campbell and A. Zujev, Some almost partition theoretic identities, Preprint, 2016.
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000712, A061159, A061161.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(add(
      d/3, 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/(3n)) 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/(3*n))*Sum_{k=1..n} c(n-k)*sigma(k), n>0, c(0)=1.