0,2

Row sums are:n! ;

{1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600,...}.

These polynomials are hit polynomials for the reduced menage problem from Riordan.

This first version didn't check with Riordans's table :

I used x^m instead of (x-1)^m.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 197-199

p(x,n)=If[n == 0, 1, Sum[Binomial[2*n - m, m]*(n - m)!*(2*n/(2*n - m))*(x - 1)^m, {m, 0, n}]];

t(n,m)=coefficients(p(x,n))

{1},

{-1, 2},

{0, 0, 2},

{1, 0, 3, 2},

{2, 8, 4, 8, 2},

{13, 30, 40, 20, 15, 2},

{80, 192, 210, 152, 60, 24, 2},

{579, 1344, 1477, 994, 469, 140, 35, 2},

{4738, 10800, 11672, 7888, 3660, 1232, 280, 48, 2},

{43387, 97434, 104256, 70152, 32958, 11268, 2856, 504, 63, 2},

{439792, 976000, 1036050, 695760, 328920, 115056, 30300, 6000, 840, 80, 2},

{4890741, 10749024, 11338855, 7603266, 3614490, 1284360, 349734, 73260, 11649, 1320, 99, 2},

{59216642, 129103992, 135494844, 90758872, 43341822, 15596208, 4351368, 951984, 162558, 21208, 1980, 120, 2}

Table[CoefficientList[If[n == 0, 1, Sum[Binomial[2*n - m, m]*(n - m)!*(2*n/(2*n - m))*(x - 1)^m, {m, 0, n}]], x], {n, 0, 12}];

Flatten[%]

Cf. A094314. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 24 2009]

Sequence in context: A143620 A025843 A035437 this_sequence A029304 A030202 A159818

Adjacent sequences: A156993 A156994 A156995 this_sequence A156997 A156998 A156999

sign,tabl,uned

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 20 2009