A176666 A triangle of polynomial coefficients:p(x,n)=Sum[(2*k + 1)^n*k!*Binomial[x, k], {k, 0, n}].

1, 1, 3, 1, -16, 25, 1, 588, -904, 343, 1, -35108, 65593, -36965, 6561, 1, 3541662, -7450307, 5299298, -1551461, 161051, 1, -539667860, 1239476145, -1027098387, 393094596, -70630574, 4826809, 1, 115929493398, -285126982237, 264011385389

Offset: 0

Author

Roger L. Bagula, Apr 23 2010

Keywords

• sign
• tabl
• uned

Comments

Row sums are:A103457;

{1, 4, 10, 28, 82, 244, 730, 2188, 6562, 19684, 59050,...}.

Examples

			{1},
{1, 3},
{1, -16, 25},
{1, 588, -904, 343},
{1, -35108, 65593, -36965, 6561},
{1, 3541662, -7450307, 5299298, -1551461, 161051},
{ 1, -539667860, 1239476145, -1027098387, 393094596, -70630574, 4826809},
{1, 115929493398, -285126982237, 264011385389, -120438105421, 28978650041, -3525298358, 170859375},
{1, -33405526460804, 86851508060145, -87619801707127, 45402414077950, -13236000326919, 2193188923598, -192758317723, 6975757441},
{1, 12439546100725062, -33876724511327305, 36619991865553925, -20936375400104384, 7018154767854372, -1426322806941012, 172905465804793, -11498169243547, 322687697779},
{1, -5815351979718349460, 16476663041157314889, -18861838035155184791, 11671607490973992658, -4358525114199083475, 1028212770824559839, -154333184246062051, 14292794059654483, -744463577761244, 16679880978201}
		

Crossrefs

Cf. A103457

Programs

Mathematica

Clear[p, x, n]
p[x_, n_] := Sum[(2*k + 1)^n*k!*Binomial[x, k], {k, 0, n}];
Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}];
Flatten[%]

Formula

p(x,n)=Sum[(2*k + 1)^n*k!*Binomial[x, k], {k, 0, n}];

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