A147290 A Pascal triangle with an Eulerian-number shift: p(x,n)=If[n < 1, (x + 1)^(n + 1), (x + 1)^(n + 1) + (1 - x)^(n + 1)*PolyLog[ -n, x]].

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 5, 10, 5, 1, 1, 6, 21, 21, 6, 1, 1, 7, 41, 86, 41, 7, 1, 1, 8, 78, 337, 337, 78, 8, 1, 1, 9, 148, 1247, 2486, 1247, 148, 9, 1, 1, 10, 283, 4377, 15745, 15745, 4377, 283, 10, 1, 1, 11, 547, 14728, 88444, 156442, 88444, 14728, 547, 11, 1, 1

Offset

-1,5

Comments

Row sums are {1, 2, 5, 10, 22, 56, 184, 848, 5296, 40832, 363904, 3630848, ...}.

Links

Table of n, a(n) for n=-1..65.

Formula

p(x,n)=If[n < 1, (x + 1)^(n + 1), (x + 1)^(n + 1) + (1 - x)^(n + 1)*PolyLog[ -n, x]]; t(n,m)=Coefficients(p(x,n)0.

Example

{1}, {1, 1}, {1, 3, 1}, {1, 4, 4, 1}, {1, 5, 10, 5, 1}, {1, 6, 21, 21, 6, 1}, \ {1, 7, 41, 86, 41, 7, 1}, {1, 8, 78, 337, 337, 78, 8, 1}, {1, 9, 148, 1247, 2486, 1247, 148, 9, 1}, {1, 10, 283, 4377, 15745, 15745, 4377, 283, 10, 1}, {1, 11, 547, 14728, 88444, 156442, 88444, 14728, 547, 11, 1}, {1, 12, 1068, 48005, 455522, 1310816, 1310816, 455522, 48005, 1068, 12, 1}

Mathematica

Clear[p, n, m]; p[x_, n_] = If[n < 1, (x + 1)^(n + 1), (x + 1)^(n + 1) + (1 - x)^(n + 1)*PolyLog[ -n, x]]; Table[FullSimplify[ExpandAll[p[x, n]]], {n, -1, 10}]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, -1, 10}]; Flatten[%]

Crossrefs

Sequence in context: A173117 A050177 A013580 * A026670 A131402 A238498

Adjacent sequences: A147287 A147288 A147289 * A147291 A147292 A147293

Keyword

nonn,tabl

Author

Roger L. Bagula, Nov 04 2008

Extensions

Edited by N. J. A. Sloane, Nov 07 2008

Status

approved