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A002539
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Eulerian numbers of the second kind read by diagonals <<n+3, n>>
(Formerly M5126 N2221)
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6
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1, 22, 328, 4400, 58140, 785304, 11026296, 162186912, 2507481216, 40788301824, 697929436800, 12550904017920, 236908271543040, 4687098165573120
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Eulerian permutations of the multiset {1,1,2,2,...,n+3,n+3} with n ascents.Eulerian permutations have the restriction that for all m, all integers between the two copies of m are less than m. In particular, the two 1s are always next to each other.
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REFERENCES
| I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, A 24 (1978), 24-33.
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2nd edition; Addison-Wesley, 1994, pp270-271.
O. J. Munch, Om potensproduktsummer [ Norwegian, English summary ], Nordisk Matematisk Tidskrift, 7 (1959), 5-19.
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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LINKS
| T. D. Noe, Table of n, a(n) for n=1..100
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FORMULA
| a(n)= (n+4)*a(n-1) + n*A002538(n), n>=2. a(1)=1.
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EXAMPLE
| For instance, a(1) = 22 because among the 7!! = 105 permutations of {1,1,2,2,3,3,4,4} selected according to the definition of Eulerian numbers of the second kind, only 22 contain n = 1 descent, namely :
11223443, 11224433, 11233244, 11233442, 11244233,
11332244, 11334422, 11442233, 12213344, 12233144,
12233441, 12244133, 13312244, 13344122, 14412233,
22113344, 22331144, 22334411, 22441133, 33112244,
33441122, 44112233 (Jean-François Alcover - Mar 28 2011).
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MATHEMATICA
| a[1]=1; a[2]=22; a[n_] := a[n] = ((n-1)*(n-1)!*n^3 - (n+2)*(n+3)*a[n-2]*n + (n*(2*n+5)-4)*a[n-1]) / (n-1); Table[a[n], {n, 14}] (* Jean-François Alcover, Mar 23 2011 *)
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CROSSREFS
| 3rd diagonal of A008517, third column of A112007.
Sequence in context: A025941 A125479 A047868 * A019958 A021644 A021834
Adjacent sequences: A002536 A002537 A002538 * A002540 A002541 A002542
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe, Robert G. Wilson v (rgwv(AT)rgwv.com), Mira Bernstein (mira(AT)math.berkeley.edu)
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