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%I M1194 #66 Dec 26 2021 21:05:34
%S 0,1,2,4,9,24,76,279,1156,5296,26443,142418,820988,5034585,32679022,
%T 223578344,1606536889,12086679036,94951548840,777028354999,
%U 6609770560056,58333928795428,533203744952179,5039919483399502,49191925338483848,495150794633289137
%N a(0) = 0; for n>0, a(n) = Sum_k={0..n-1} Bell(k), where the Bell numbers Bell(k) are given in A000110.
%C Counts rhyme schemes.
%C Row sums of triangle A137596 starting with offset 1. - _Gary W. Adamson_, Jan 29 2008
%C 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 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 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A005001/b005001.txt">Table of n, a(n) for n = 0..100</a>
%H J. Riordan, <a href="/A005000/a005000.pdf">Cached copy of paper</a>
%H 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 a(0) = 0; for n >= 0, a(n+1) = 1 + Sum_{j=1..n} (C(n, j)-C(n, j+1))*a(j).
%F a(n) = A000110(n) - A171859(n). - _Emeric Deutsch_, May 01 2010
%F 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 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 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 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 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 E.g.f. A(x) satisfies: A'(x) = A(x) + exp(exp(x)-1). - _Geoffrey Critzer_, Feb 04 2014
%F G.f.: (x/(1 - x)) * Sum_{i>=0} x^i / Product_{j=1..i} (1 - j*x). - _Ilya Gutkovskiy_, Jun 05 2017
%F a(n) ~ Bell(n) / (n/LambertW(n) - 1). - _Vaclav Kotesovec_, Jul 28 2021
%p with(combinat): seq(add(bell(j), j = 0 .. n-1), n = 0 .. 22); # _Emeric Deutsch_, May 01 2010
%t 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 BellB /@ Range[0, 30] // Accumulate // Prepend[#, 0]& (* _Jean-François Alcover_, Oct 19 2019 *)
%o (Python)
%o # Python 3.2 or higher required.
%o from itertools import accumulate
%o A005001_list, blist, a, b = [0,1,2], [1], 2, 1
%o for _ in range(30):
%o ....blist = list(accumulate([b]+blist))
%o ....b = blist[-1]
%o ....a += b
%o ....A005001_list.append(a) # _Chai Wah Wu_, Sep 19 2014
%Y Partial sums of A000110, partial sums give A029761.
%Y Equals A024716(n-1) + 1.
%Y Cf. A102735, A094262, A000110, A008277, A102639, A003422, A000166, A000204, A000045, A000108.
%Y Cf. A137596.
%Y Cf. A171859. - _Emeric Deutsch_, May 01 2010
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
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