A173109 Row sums of triangle A173108.

1, 1, 3, 6, 18, 58, 221, 935, 4361, 22082, 120336, 700652, 4333933, 28345089, 195233255, 1411303634, 10675375402, 84276173438, 692752181561, 5917018378495, 52416910416933, 480786834535246, 4559132648864256, 44632792689619592, 450518001943669545

Offset: 0

Author

Gary W. Adamson, Feb 09 2010

Keywords

• nonn

Examples

			a(5) = 58 = 52 + 5 + 1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..576

Crossrefs

Cf. A000110, A087650, A173108.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> add(b(n-2*k), k=0..iquo(n, 2)):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 17 2021

Mathematica

a[n_] := Sum[BellB[n-2k], {k, 0, Quotient[n, 2]}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 22 2022 *)

Formula

G.f.: G(0)/(1-x^2)/(1+x) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k-1) - x*(2*k+1)*(2*k+3)*(2*x*k-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k+x-1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 04 2013.

G.f.: ( G(0) - 1 )/(1-x^2) where G(k) = 1 + (1-x)/(1-x*k)/(1-x/(x+(1-x)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 22 2013

a(n+1) - a(n) = A087650(n+1). - Vladimir Reshetnikov, Oct 29 2015

a(n) = Sum_{k=0..floor(n/2)} A000110(n-2*k). - Alois P. Heinz, Jun 17 2021

Extensions

More terms from Sergei N. Gladkovskii, Jan 04 2013