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

%I A005001 M1194 #76 Mar 15 2025 05:28:23
%S A005001 0,1,2,4,9,24,76,279,1156,5296,26443,142418,820988,5034585,32679022,
%T A005001 223578344,1606536889,12086679036,94951548840,777028354999,
%U A005001 6609770560056,58333928795428,533203744952179,5039919483399502,49191925338483848,495150794633289137
%N A005001 a(n) = Sum_{k=0..n-1} Bell(k), where the Bell numbers Bell(k) are given in A000110.
%C A005001 Counts rhyme schemes.
%C A005001 Row sums of triangle A137596 starting with offset 1. - _Gary W. Adamson_, Jan 29 2008
%C A005001 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
%C A005001 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
%D A005001 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005001 T. D. Noe, <a href="/A005001/b005001.txt">Table of n, a(n) for n = 0..100</a>
%H A005001 J. Riordan, <a href="/A005000/a005000.pdf">Cached copy of paper</a>
%H A005001 J. Riordan, <a href="http://dx.doi.org/10.1111/j.1749-6632.1979.tb32823.x">A budget of rhyme scheme counts</a>, 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.
%F A005001 a(0) = 0; for n >= 0, a(n+1) = 1 + Sum_{j=1..n} (C(n, j)-C(n, j+1))*a(j).
%F A005001 a(n) = A000110(n) - A171859(n). - _Emeric Deutsch_, May 01 2010
%F A005001 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
%F A005001 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
%F A005001 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
%F A005001 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
%F A005001 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
%F A005001 E.g.f. A(x) satisfies: A'(x) = A(x) + exp(exp(x)-1). - _Geoffrey Critzer_, Feb 04 2014
%F A005001 G.f.: (x/(1 - x)) * Sum_{i>=0} x^i / Product_{j=1..i} (1 - j*x). - _Ilya Gutkovskiy_, Jun 05 2017
%F A005001 a(n) ~ Bell(n) / (n/LambertW(n) - 1). - _Vaclav Kotesovec_, Jul 28 2021
%p A005001 with(combinat): seq(add(bell(j), j = 0 .. n-1), n = 0 .. 22); # _Emeric Deutsch_, May 01 2010
%t A005001 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 *)
%t A005001 BellB /@ Range[0, 30] // Accumulate // Prepend[#, 0]& (* _Jean-François Alcover_, Oct 19 2019 *)
%o A005001 (Python)
%o A005001 # Python 3.2 or higher required.
%o A005001 from itertools import accumulate
%o A005001 A005001_list, blist, a, b = [0,1,2], [1], 2, 1
%o A005001 for _ in range(30):
%o A005001     blist = list(accumulate([b]+blist))
%o A005001     b = blist[-1]
%o A005001     a += b
%o A005001     A005001_list.append(a) # _Chai Wah Wu_, Sep 19 2014
%Y A005001 Partial sums of A000110, partial sums give A029761.
%Y A005001 Equals A024716(n-1) + 1.
%Y A005001 Cf. A102735, A094262, A000110, A008277, A102639, A003422, A000166, A000204, A000045, A000108.
%Y A005001 Cf. A137596.
%Y A005001 Cf. A171859.
%K A005001 nonn,easy
%O A005001 0,3
%A A005001 _N. J. A. Sloane_