A174859 A triangle sequence of polynomial coefficients:p(x,n)=Sum[Binomial[n, k]*(-x)^k*Sum[StirlingS2[n, m]*x^m, {m, 0, n - k}], {k, 0, n}].

1, 0, 1, 0, 1, -1, 0, 1, 0, -5, 0, 1, 3, -16, 15, 0, 1, 10, -40, 25, 56, 0, 1, 25, -81, -30, 370, -455, 0, 1, 56, -119, -469, 1841, -1960, -237, 0, 1, 119, -22, -2527, 7448, -5768, -7420, 16947, 0, 1, 246, 766, -10359, 24627, -2289, -76692, 126504, -64220, 0, 1

Offset: 0

Author

Roger L. Bagula, Mar 31 2010

Keywords

• sign
• tabl
• uned

Comments

Row sums are:

{1, 1, 0, -4, 3, 52, -170, -887, 8778, -1416, -415734,...}.

Examples

			{1},
{0, 1},
{0, 1, -1},
{0, 1, 0, -5},
{0, 1, 3, -16, 15},
{0, 1, 10, -40, 25, 56},
{0, 1, 25, -81, -30, 370, -455},
{0, 1, 56, -119, -469, 1841, -1960, -237},
{0, 1, 119, -22, -2527, 7448, -5768, -7420, 16947},
{0, 1, 246, 766, -10359, 24627, -2289, -76692, 126504, -64220},
{0, 1, 501, 4265, -36320, 60215, 119760, -570627, 784245, -248280, -529494}
		

References

• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 77.

Crossrefs

Cf. A008299

Programs

Mathematica

Clear[p, x, n];
p[x_, n_] = Sum[Binomial[n, k]*(-x)^k*Sum[StirlingS2[n, m]*x^m, {m, 0, n - k}], {k, 0, n}];
Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[%]

Formula

p(x,n)=Sum[Binomial[n, k]*(-x)^k*Sum[StirlingS2[n, m]*x^m, {m, 0, n - k}], {k, 0, n}];

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