A174833 A triangular sequence of polynomial coefficients:p(x,n)=Sum[Eulerian[n + 1, k]*Product[x + i, {i, 0, n - k + 1}]*(-x)^k, {k, 0, n}]/x

1, 2, 2, 6, 3, -5, -2, 24, -16, -64, -24, 120, -350, -679, -151, 74, 16, 720, -5076, -6746, 1198, 2692, 544, 5040, -73332, -55628, 80239, 68081, 8858, -2000, -272, 40320, -1135296, -18712, 2522252, 1417588, -83312, -155392, -19840, 362880

Offset

0,2

Comments

Row sums are:

{1, 4, 2, -80, -970, -6668, 30986, 2567608, 67974542, 1084990396, -6362961886,...}

Links

Table of n, a(n) for n=0..39.

Formula

p(x,n)=Sum[Eulerian[n + 1, k]*Product[x + i, {i, 0, n - k + 1}]*(-x)^k, {k, 0, n}]/x

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

Example

{1},
{2, 2},
{6, 3, -5, -2},
{24, -16, -64, -24},
{120, -350, -679, -151, 74, 16},
{720, -5076, -6746, 1198, 2692, 544},
{5040, -73332, -55628, 80239, 68081, 8858, -2000, -272},
{40320, -1135296, -18712, 2522252, 1417588, -83312, -155392, -19840},
{362880, -19214064, 19785852, 68794296, 21423037, -14938145, -7669234, -662752, 84736, 7936},
{3628800, -356968800, 901740888, 1768795280, -83612926, -876416978, -291269756, 6024320, 12008192, 1061376},
{39916800, -7267692960, 33899921496, 41987822092, -30465270802, -40831787105, -7672559733, 2867304822, 1021675584, 63226000, -5164288, -353792}

Mathematica

<< DiscreteMath`Combinatorica`
Clear[p, n, x, k];
p[x, 0] := 1;
p[x_, n_] := Sum[Eulerian[n + 1, k]* Product[x + i, {i, 0, n - k + 1}]*(-x)^k, {k, 0, n}]/x;
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
Flatten[%]

Crossrefs

Cf. A174789

Sequence in context: A277011 A277021 A275037 * A085738 A227608 A276484

Adjacent sequences: A174830 A174831 A174832 * A174834 A174835 A174836

Keyword

sign,tabl,uned

Author

Roger L. Bagula, Mar 30 2010

Status

approved