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, 1, 1, 1013

Offset

1,2

Comments

Not entirely symmetrical, the x^(n+1)+1 polynomials was added to remove zeros and make the triangle more symmetrical.

Row sums are:

{1, 3, 3, 4, 8, 26, 122, 722, 5042, 40322, 362882, 3628802}.

Links

Table of n, a(n) for n=1..69.

Formula

f(x,n)=x^(n+1)+1+A123125(x,n).

Mathematica

lear[f, x, n, a] 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]

Crossrefs

Cf. A123125.

Sequence in context: A300362 A248145 A171398 * A351352 A082586 A160094

Adjacent sequences: A113604 A113605 A113606 * A113608 A113609 A113610

Keyword

nonn,tabl

Author

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

Status

approved