A113607 An extended triangle of Eulerian coefficients A123125: f(x,n)=x^(n+1)+1+A123125(x,n).

1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 11, 11, 1, 1, 1, 1, 26, 66, 26, 1, 1, 1, 1, 57, 302, 302, 57, 1, 1, 1, 1, 120, 1191, 2416, 1191, 120, 1, 1, 1, 1, 247, 4293, 15619, 15619, 4293, 247, 1, 1, 1, 1, 502, 14608, 88234, 156190, 88234, 14608, 502, 1, 1

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Aug 25 2008

Keywords

• nonn
• tabl
• less

Examples

			Triangle starts:
 1,
 2, 1,
 1, 1, 1,
 1, 1, 1, 1,
 1, 1, 4, 1, 1,
 1, 1, 11, 11, 1, 1,
 1, 1, 26, 66, 26, 1, 1,
 1, 1, 57, 302, 302, 57, 1, 1,
 1, 1, 120, 1191, 2416, 1191, 120, 1, 1,
 1, 1, 247, 4293, 15619, 15619, 4293, 247, 1, 1,
 1, 1, 502, 14608, 88234, 156190, 88234, 14608, 502, 1, 1,
 ...
		

Crossrefs

Cf. A123125.

Programs

Mathematica

f[x_, n_] := f[x,n] = x^(n + 1) + (1 - x)^(n + 1)*Sum[k^n*x^k, {k, 0, Infinity}] + 1;
Table[FullSimplify[ExpandAll[f[x, n]]], {n, 0, 10}];
a = Join[{{1}}, Table[CoefficientList[FullSimplify[ExpandAll[f[x, n]]], x], {n, 0, 10}]];
Flatten[a]