A160181 Number of partitions of sets containing from 0 to n elements into blocks of at least 2 elements.

1, 1, 2, 3, 7, 18, 59, 221, 936, 4361, 22083, 120336, 700653, 4333933, 28345090, 195233255, 1411303635, 10675375402, 84276173439, 692752181561, 5917018378496, 52416910416933, 480786834535247, 4559132648864256

Offset: 0

Author

Anonymous, May 03 2009

Keywords

• easy
• nonn

Comments

Partial sums of A000296.

a(n) is the total number of complete rhyme schemes for 0 to n lines; in other words, a(n) is the total number of rhyme schemes for 0 to n lines where each line rhymes with at least one other line.

If the restriction that the blocks of the partitions must have at least 2 elements is removed, then A005001 is obtained except for the first term of A005001.

Crossrefs

Cf. A000296, A005001, A000110.

Programs

Mathematica

m=30; CoefficientList[Series[(1+x*Sum[x^k/Product[1-p*x, {p,0,k}], {k,0,m}])/(1-x^2), {x, 0,m}], x] (* Georg Fischer, Aug 28 2020 *)

Formula

G.f.: (G(0)-1)/(1-x) where G(k) = 1 + (1-x)/(1+x-x*k)/(1-x/(x+(1-x)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 21 2013

G.f.: T(0)/(1-x), where T(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x*k)*(1-x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 19 2013

G.f.: (1+x*sum{k>=0, x^k/prod[p=0..k, 1-p*x]})/(1-x^2). - Sergei N. Gladkovskii, Jan 25 2014

Extensions

a(22)-a(23) corrected by Georg Fischer, Aug 28 2020