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A079267
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d(n,s) = number of perfect matchings on {1, 2, ..., n} with k short pairs.
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2
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1, 0, 1, 1, 1, 1, 5, 6, 3, 1, 36, 41, 21, 6, 1, 329, 365, 185, 55, 10, 1, 3655, 3984, 2010, 610, 120, 15, 1, 47844, 51499, 25914, 7980, 1645, 231, 21, 1, 721315, 769159, 386407, 120274, 25585, 3850, 406, 28, 1, 12310199, 13031514, 6539679, 2052309, 446544
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,7
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COMMENTS
| Read backwards, the n-th row of the triangle gives the Hilbert series of the variety of slopes determined by n points in the plane.
Contribution from Paul Barry (pbarry(AT)wit.ie), Nov 25 2009: (Start)
Reversal of coefficient array for the polynomials P(n,x)=sum{k=0..n, (C(n+k,2k)(2k)!/(2^k*k!))*x^k*(1-x)^(n-k)}.
Note that P(n,x)=sum{k=0..n, A001498(n,k)*x^k*(1-x)^(n-k)}. (End)
Equivalent to the original definition: Triangle of fixed-point free involutions on [1..2n] (=A001147) by number of cycles with adjacent integers. [From Olivier Gerard, Mar 23 2011]
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REFERENCES
| G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble fini totalement ordonne, Publications de l'Institut de Statistique de l'Universit\'{e} de Paris, 23 (1978), 57-74
J. L. Martin, The slopes determined by n points in the plane, preprint, 2003.
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LINKS
| J. L. Martin, The slopes determined by n points in the plane.
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FORMULA
| d(n, s) = 1/s! * sum(((-1)^(h-s)*(2*n-h)!/(2^(n-h)*(n-h)!*(h-s)!)), h=s..n)
E.g.f.: exp((x-1)*(1-sqrt(1-2*y)))/sqrt(1-2*y). [From Vladeta Jovovic (vladeta(AT)eunet.yu), Dec 15 2008]
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EXAMPLE
| Triangle begins:
1
0 1
1 1 1
5 6 3 1
36 41 21 6 1
Contribution from Paul Barry (pbarry(AT)wit.ie), Nov 25 2009: (Start)
Production matrix begins
0, 1,
1, 1, 1,
4, 4, 2, 1,
18, 18, 9, 3, 1,
96, 96, 48, 16, 4, 1,
600, 600, 300, 100, 25, 5, 1,
4320, 4320, 2160, 720, 180, 36, 6, 1,
35280, 35280, 17640, 5880, 1470, 294, 49, 7, 1,
322560, 322560, 161280, 53760, 13440, 2688, 448, 64, 8, 1
Complete this by adding top row (1,0,0,0,....) and take inverse: we obtain
1,
0, 1,
-1, -1, 1,
-2, -2, -2, 1,
-3, -3, -3, -3, 1,
-4, -4, -4, -4, -4, 1,
-5, -5, -5, -5, -5, -5, 1,
-6, -6, -6, -6, -6, -6, -6, 1,
-7, -7, -7, -7, -7, -7, -7, -7, 1,
-8, -8, -8, -8, -8, -8, -8, -8, -8, 1 (End)
The 6 involutions with no fixed point on [1..6] with only one 2-cycle with adjacent integers are ((1, 2), (3, 5), (4, 6)), ((1, 3), (2, 4), (5, 6)), ((1, 3), (2, 6), (4, 5)), ((1, 5), (2, 3), (4, 6)), ((1, 5), (2, 6), (3, 4)), and ((1, 6), (2, 5), (3, 4)).
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MAPLE
| d := (n, s) -> 1/s! * sum('((-1)^(h-s)*(2*n-h)!/(2^(n-h)*(n-h)!*(h-s)!))', 'h'=s..n):
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MATHEMATICA
| nmax = 9; d[n_, s_] := (2^(s-n)*(2n-s)!* Hypergeometric1F1[s-n, s-2n, -2])/ (s!*(n-s)!); Flatten[ Table[d[n, s], {n, 0, nmax}, {s, 0, n}]] (* From Jean-François Alcover, Oct 19 2011, after Maple *)
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CROSSREFS
| Row sums are A001147. Columns are A000806, A006198, A006199, A006200.
Sequence in context: A011499 A106599 A195448 * A060296 A114598 A199666
Adjacent sequences: A079264 A079265 A079266 * A079268 A079269 A079270
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KEYWORD
| easy,nice,nonn,tabl
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AUTHOR
| Jeremy Martin (martin(AT)math.umn.edu), Feb 05 2003
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EXTENSIONS
| Extra terms added. Paul Barry (pbarry(AT)wit.ie), Nov 25 2009
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