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A000274
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Number of permutations of length n by rises.
(Formerly M3048 N1236)
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7
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1, 3, 18, 110, 795, 6489, 59332, 600732, 6674805, 80765135, 1057289046, 14890154058, 224497707343, 3607998868005, 61576514013960, 1112225784377144, 21197714949305577, 425131949816628507, 8950146311929021210
(list; graph; refs; listen; history; internal format)
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OFFSET
| 3,2
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COMMENTS
| Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), May 25 2009: (Start)
a(n)=number of excedances in all derangements of [n-1]. Example: a(5)=18 because the derangements of {1,2,3,4} are 4*123, 3*14*2, 3*4*12, 4*3*12, 2*14*3, 2*4*13, 2*3*4*1, 3*4*21, 4*3*21 with the 18 excedances marked. An excedance of a permutation p is a position i such that p(i)>i.
a(n)=Sum(k*A046739(n,k), k>=1).
(End)
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REFERENCES
| F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
R. Mantaci and F. Rakotondrajao, Exceedingly deranging!, Advances in Appl. Math., 30 (2003), 177-188. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), May 25 2009]
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210 (divided by 2).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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FORMULA
| a(n) = (1 + n) a(n - 1) + (3 + n) a(n - 2) + (3 - n) a(n - 3) + (2 - n) a(n - 4).
E.g.f.: x^2/2*exp(-x)/(1-x)^2. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 03 2003
a(n)=(n-1)^2/(n-2)*a(n-1)-(-1)^n*(n-1)/2, n>2, a(2)=0. - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 31 2003
(1/2){[n!/e] - [(n-1)!/e]} (conjectured).
a(n) = (n-1)*GAMMA(n,-1)*exp(-1)/2 where GAMMA = incomplete Gamma function [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Nov 11 2009]
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MAPLE
| a:=n->sum(n!*sum((-1)^k/k!/2, j=1..n), k=0..n): seq(a(n), n=2..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 17 2007
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MATHEMATICA
| Table[Subfactorial[n]*n/2, {n, 2, 20}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2009]
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CROSSREFS
| Cf. A010027, A000255, A000166, A000313, A001260, A001261.
A diagonal in triangle A010027.
A046739 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), May 25 2009]
Sequence in context: A074571 A114311 A134092 * A193236 A199259 A163471
Adjacent sequences: A000271 A000272 A000273 * A000275 A000276 A000277
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KEYWORD
| easy,nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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