A038561 Left-hand border of triangle A046937.

1, 2, 3, 8, 24, 83, 324, 1400, 6609, 33758, 185136, 1083233, 6726366, 44130128, 304741623, 2207682188, 16729947276, 132281116715, 1088831511000, 9311082630620, 82569723552561, 758057178490082, 7194283782101844, 70481938088367569

Offset: 0

Author

Henry Gould, N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

For n>1: a(n) is the number of entries in the last blocks of all set partitions of [n]. a(3) = 8 because the number of entries in the last blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 3+1+1+2+1 = 8. - Alois P. Heinz, May 08 2017

References

• H. W. Gould, A linear binomial recurrence and the Bell numbers and polynomials, preprint, 1998

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..500
• R. K. Guy, Letters to N. J. A. Sloane, June-August 1968

Crossrefs

A040027(n) + B(n), where B(n) = Bell numbers A000110.

Related to A000110, A040027, A038559, A038560.

Column k=1 of A286416 (for n>1).

Programs

Haskell

a038561 = head . a046937_row  -- Reinhard Zumkeller, Jan 06 2014

Maple

A038561List := proc(m) local A, P, n; A := [1,2]; P := [1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([A[-1], op(P)]);
A := [op(A), P[-1]] od; A end: A038561List(24); # Peter Luschny, Mar 24 2022

Mathematica

a[0, 0] = 1; a[1, 0] = 2; a[n_, 0] := a[n-1, n-1]; a[n_, k_] := a[n, k] = a[n, k-1] + a[n-1, k-1]; a[n_] := a[n, 0]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jun 06 2013 *)

Formula

G.f. A(x) satisfies: A(x) = 1 + x * (1 + A(x/(1 - x)) / (1 - x)). - Ilya Gutkovskiy, Jun 30 2020