A051573 INVERTi transform of A000081 = [1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486,...].

1, 1, 1, 2, 3, 8, 16, 41, 98, 250, 631, 1646, 4285, 11338, 30135, 80791, 217673, 590010, 1606188, 4392219, 12055393, 33206321, 91752211, 254261363, 706465999, 1967743066, 5493195530, 15367129299, 43073007846, 120949992543, 340206026166, 958444631917

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A052250, A000081.

Programs

Maple

with(numtheory):
b:= proc(n) option remember; local d, j; `if` (n<2, n,
      (add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
    end:
a:= proc(n) option remember; local i; `if`(n<0, -1,
      -add(a(n-i) *b(i+1), i=1..n+1))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 17 2013

Mathematica

b[n_] := b[n] = If[n < 2, n, Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}]/(n-1)]; a[n_] := a[n] = If[n < 0, -1, -Sum[a[n-i]*b[i+1], {i, 1, n+1}]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 16 2014, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.9557652856519949747148175241..., c = A187770 = 0.4399240125710253040409033914... . - Vaclav Kotesovec, Sep 06 2014