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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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13
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Counts rhyme schemes.
Row sums of triangle A137596 starting with offset 1. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 29 2008
With offset 1 = binomial transform of the Bell numbers, A000110 starting (1, 1, 1, 2, 5, 15, 52, 203,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 04 2008]
Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), May 01 2010: (Start)
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}.
a(n)= A000110(n) - A171859(n).
(End)
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REFERENCES
| 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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
J. Riordan, Cached copy of paper
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FORMULA
| a(0) = 0; for n >= 0, a(n+1) = 1 + Sum_{j=1..n} (C(n, j)-C(n, j+1))*a(j).
Sum_{i=1..n} Bell(i) = 1 + C(n, 2) + 2*C(n-3, 1) + 8*C(n-4, 1) + C(n-3, 2) + 22*C(n-5, 1) + 13*C(n-4, 2) + 52*C(n-6, 1) + 74*C(n-5, 2) + 10*C(n-4, 3) + 114*C(n-7, 1) + 314*C(n-6, 2) + 134*C(n-5, 3) + 3*C(n-4, 4) + 240*C(n-8, 1) + 1155*C(n-7, 2) + 1024*C(n-6, 3) + 134*C(n-5, 4) + 494*C(n-9, 1) + ..... . - Andre F. Labossiere (boronali(AT)laposte.net), Feb 11 2005
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MAPLE
| with(combinat): seq(add(bell(j), j = 0 .. n-1), n = 0 .. 22); [From Emeric Deutsch (deutsch(AT)duke.poly.edu), May 01 2010]
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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.
A171859 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), May 01 2010]
Sequence in context: A009283 A125654 A141824 * A091151 A093542 A000667
Adjacent sequences: A004998 A004999 A005000 * A005002 A005003 A005004
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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