%I A079267
%S A079267 1,0,1,1,1,1,5,6,3,1,36,41,21,6,1,329,365,185,55,10,1,3655,3984,2010,
%T A079267 610,120,15,1,47844,51499,25914,7980,1645,231,21,1,721315,769159,386407,
%U A079267 120274,25585,3850,406,28,1,12310199,13031514,6539679,2052309,446544
%N A079267 d(n,s) = number of perfect matchings on {1, 2, ..., n} with k short pairs.
%C A079267 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.
%C A079267 Contribution from Paul Barry (pbarry(AT)wit.ie), Nov 25 2009: (Start)
%C A079267 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)}.
%C A079267 Note that P(n,x)=sum{k=0..n, A001498(n,k)*x^k*(1-x)^(n-k)}. (End)
%D A079267 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
%D A079267 J. L. Martin, The slopes determined by n points in the plane, preprint, 
               2003.
%H A079267 J. L. Martin, <a href="http://www.math.umn.edu/~martin/math/slopes.pdf">
               The slopes determined by n points in the plane</a>.
%F A079267 d(n, s) = 1/s! * sum(((-1)^(h-s)*(2*n-h)!/(2^(n-h)*(n-h)!*(h-s)!)), h=s..n)
%F A079267 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]
%e A079267 Triangle begins:
%e A079267 1
%e A079267 0 1
%e A079267 1 1 1
%e A079267 5 6 3 1
%e A079267 36 41 21 6 1
%e A079267 Contribution from Paul Barry (pbarry(AT)wit.ie), Nov 25 2009: (Start)
%e A079267 Production matrix begins
%e A079267 0, 1,
%e A079267 1, 1, 1,
%e A079267 4, 4, 2, 1,
%e A079267 18, 18, 9, 3, 1,
%e A079267 96, 96, 48, 16, 4, 1,
%e A079267 600, 600, 300, 100, 25, 5, 1,
%e A079267 4320, 4320, 2160, 720, 180, 36, 6, 1,
%e A079267 35280, 35280, 17640, 5880, 1470, 294, 49, 7, 1,
%e A079267 322560, 322560, 161280, 53760, 13440, 2688, 448, 64, 8, 1
%e A079267 Complete this by adding top row (1,0,0,0,....) and take inverse: we obtain
%e A079267 1,
%e A079267 0, 1,
%e A079267 -1, -1, 1,
%e A079267 -2, -2, -2, 1,
%e A079267 -3, -3, -3, -3, 1,
%e A079267 -4, -4, -4, -4, -4, 1,
%e A079267 -5, -5, -5, -5, -5, -5, 1,
%e A079267 -6, -6, -6, -6, -6, -6, -6, 1,
%e A079267 -7, -7, -7, -7, -7, -7, -7, -7, 1,
%e A079267 -8, -8, -8, -8, -8, -8, -8, -8, -8, 1 (End)
%p A079267 d := (n,s) -> 1/s! * sum('((-1)^(h-s)*(2*n-h)!/(2^(n-h)*(n-h)!*(h-s)!))',
               'h'=s..n):
%Y A079267 Row sums are A001147. Columns are A000806, A006198, A006199, A006200.
%Y A079267 Sequence in context: A018851 A011499 A106599 this_sequence A060296 A152061 
               A114598
%Y A079267 Adjacent sequences: A079264 A079265 A079266 this_sequence A079268 A079269 
               A079270
%K A079267 easy,nice,nonn,tabl,new
%O A079267 0,7
%A A079267 Jeremy Martin (martin(AT)math.umn.edu), Feb 05 2003
%E A079267 Extra terms added. Paul Barry (pbarry(AT)wit.ie), Nov 25 2009