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A005001
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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.
(Formerly M1194)
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18
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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
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OFFSET
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0,3
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COMMENTS
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Counts rhyme schemes.
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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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.
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FORMULA
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a(0) = 0; for n >= 0, a(n+1) = 1 + Sum_{j=1..n} (C(n, j)-C(n, j+1))*a(j).
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
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MAPLE
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with(combinat): seq(add(bell(j), j = 0 .. n-1), n = 0 .. 22); # Emeric Deutsch, May 01 2010
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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