0,7

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)

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.

J. L. Martin, The slopes determined by n points in the plane.

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]

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)

d := (n, s) -> 1/s! * sum('((-1)^(h-s)*(2*n-h)!/(2^(n-h)*(n-h)!*(h-s)!))', 'h'=s..n):

Row sums are A001147. Columns are A000806, A006198, A006199, A006200.

Sequence in context: A018851 A011499 A106599 this_sequence A060296 A152061 A114598

Adjacent sequences: A079264 A079265 A079266 this_sequence A079268 A079269 A079270

easy,nice,nonn,tabl,new

Jeremy Martin (martin(AT)math.umn.edu), Feb 05 2003

Extra terms added. Paul Barry (pbarry(AT)wit.ie), Nov 25 2009