A005001 a(n) = Sum_{k=0..n-1} Bell(k), where the Bell numbers Bell(k) are given in A000110.

0, 1, 2, 4, 9, 24, 76, 279, 1156, 5296, 26443, 142418, 820988, 5034585, 32679022, 223578344, 1606536889, 12086679036, 94951548840, 777028354999, 6609770560056, 58333928795428, 533203744952179, 5039919483399502, 49191925338483848, 495150794633289137

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Counts rhyme schemes.

Row sums of triangle A137596 starting with offset 1. - Gary W. Adamson, Jan 29 2008

With offset 1 = binomial transform of the Bell numbers, A000110 starting (1, 1, 1, 2, 5, 15, 52, 203, ...). - Gary W. Adamson, Dec 04 2008

a(n) is the number of partitions of the set {1,2,...,n} in which n is either a singleton or it is in a block of consecutive integers. Example: a(3)=4 because we have 123, 1-23, 12-3, and 1-2-3. Deleting the blocks containing n=3, we obtain: empty, 1, 12, 1-2, i.e., all the partitions of the sets: empty, {1}, and {1,2}. - Emeric Deutsch, May 01 2010

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..100
• J. Riordan, Cached copy of paper
• J. Riordan, A budget of rhyme scheme counts, pp. 455 - 465 of Second International Conference on Combinatorial Mathematics, New York, 1978. Edited by Allan Gewirtz and Louis V. Quintas. Annals New York Academy of Sciences, 319, 1979.

Crossrefs

Partial sums of A000110, partial sums give A029761.

Equals A024716(n-1) + 1.

Cf. A102735, A094262, A000110, A008277, A102639, A003422, A000166, A000204, A000045, A000108.

Cf. A137596.

Cf. A171859.

Programs

Maple

with(combinat): seq(add(bell(j), j = 0 .. n-1), n = 0 .. 22); # Emeric Deutsch, May 01 2010

Mathematica

nn=20;Range[0,nn]!CoefficientList[Series[Exp[-1](-Exp[Exp[x]]+Exp[1+x]-Exp[x]ExpIntegralEi[1]+Exp[x]ExpIntegralEi[Exp[x]]),{x,0,nn}],x] (* Geoffrey Critzer, Feb 04 2014 *)
BellB /@ Range[0, 30] // Accumulate // Prepend[#, 0]& (* Jean-François Alcover, Oct 19 2019 *)

Python

# Python 3.2 or higher required.
from itertools import accumulate
A005001_list, blist, a, b = [0,1,2], [1], 2, 1
for _ in range(30):
    blist = list(accumulate([b]+blist))
    b = blist[-1]
    a += b
    A005001_list.append(a) # Chai Wah Wu, Sep 19 2014

Formula

a(0) = 0; for n >= 0, a(n+1) = 1 + Sum_{j=1..n} (C(n, j)-C(n, j+1))*a(j).

a(n) = A000110(n) - A171859(n). - Emeric Deutsch, May 01 2010

G.f.: x*( 1 + (G(0)+1)*x/(1-x) ) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k+x-1) - x*(2*k+1)*(2*k+3)*(2*x*k+x-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k+2*x-1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 20 2012

G.f.: x*G(0)/(1-x^2) 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) )); (continued fraction). - Sergei N. Gladkovskii, Dec 22 2012

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

G.f.: (G(0)-1)*x/(1-x^2) where G(k) = 1 + 1/(1-k*x)/(1-x/(x+1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Feb 06 2013

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

E.g.f. A(x) satisfies: A'(x) = A(x) + exp(exp(x)-1). - Geoffrey Critzer, Feb 04 2014

G.f.: (x/(1 - x)) * Sum_{i>=0} x^i / Product_{j=1..i} (1 - j*x). - Ilya Gutkovskiy, Jun 05 2017

a(n) ~ Bell(n) / (n/LambertW(n) - 1). - Vaclav Kotesovec, Jul 28 2021