0,8

Row sums are one.

Second column is negative Eulerian numbers.

p(x,n)=p[x_, n_] = ((1 + x - x^3)^ (n + 1))*Sum[(k + 1)^n*(-x + x^3)^k, {k, 0, Infinity}];

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

{1},

{1},

{1, -1, 0, 1},

{1, -4, 1, 4, -2, 0, 1},

{1, -11, 11, 10, -22, 3, 11, -3, 0, 1},

{1, -26, 66, 0, -131, 78, 62, -78, 6, 26, -4, 0, 1},

{1, -57, 302, -245, -547, 905, 74, -901, 342, 292, -228, 10, 57, -5, 0, 1},

{1, -120, 1191, -2296, -1191, 7128, -3572, -6648, 7140, 1216, -4749, 1200, 1171, -600, 15, 120, -6, 0, 1},

{1, -247, 4293, -15372, 7033, 42564, -57936, -25393, 92232, -27304, -58771, 42909, 10679, -21430, 3705, 4258, -1482, 21, 247, -7, 0, 1},

{1, -502, 14608, -87732, 126974, 176468, -595544, 175966, 849493, -790592, -405648, 871798, -135942, -423600, 219064, 70664, -87578, 10542, 14552, -3514, 28, 502, -8, 0, 1},

{1, -1013, 47840, -454179, 1214674, 55222, -4738384, 5138354, 5131985, -12313469, 1578360, 12098909, -7765122, -4877406, 6771152, -363962, -2660242, 1004514, 398464, -334754, 28364, 47756, -8104, 36, 1013, -9, 0, 1}

Clear[p, x, n, m];

p[x_, n_] = ((1 + x - x^3)^ (n + 1))*Sum[(k + 1)^n*(-x + x^3)^k, {k, 0, Infinity}];

Table[Expand[FullSimplify[ExpandAll[p[x, n]]]], {n, 0, 10}];

Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];

Flatten[%]

Adjacent sequences: A156893 A156894 A156895 this_sequence A156897 A156898 A156899

Sequence in context: A097936 A050338 A077088 this_sequence A002193 A020807 A055190

tabl,uned,sign

Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 17 2009