A147532 Shifted Pascal sequence: p(x,n)=(1 + x)^(n + 1) + If[n < 2, 0, x*((1 - x)^(n + 1)*PolyLog[ -n, x]/x + (1 + x)^(n - 1))/2].

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 5, 9, 5, 1, 1, 6, 17, 17, 6, 1, 1, 7, 30, 56, 30, 7, 1, 1, 8, 52, 191, 191, 52, 8, 1, 1, 9, 91, 659, 1288, 659, 91, 9, 1, 1, 10, 163, 2241, 7953, 7953, 2241, 163, 10, 1, 1, 11, 300, 7438, 44355, 78382, 44355, 7438, 300, 11, 1, 1, 12, 566, 24103

Offset

-1,5

Comments

Row sums are: {1, 2, 4, 10, 21, 48, 132, 504, 2808, 20736, 182592, 1816704}.

Links

Table of n, a(n) for n=-1..68.

Formula

p(x,n)=(1 + x)^(n + 1) + If[n < 2, 0, x*((1 - x)^(n + 1)*PolyLog[ -n, x]/x + (1 + x)^(n - 1))/2]; t(n,m)=coefficients(p(x,n)).

Example

{1}, {1, 1}, {1, 2, 1}, {1, 4, 4, 1}, {1, 5, 9, 5, 1}, {1, 6, 17, 17, 6, 1}, {1, 7, 30, 56, 30, 7, 1}, {1, 8, 52, 191, 191, 52, 8, 1}, {1, 9, 91, 659, 1288, 659, 91, 9, 1}, {1, 10, 163, 2241, 7953, 7953, 2241, 163, 10, 1}, {1, 11, 300, 7438, 44355, 78382, 44355, 7438, 300, 11, 1}, {1, 12, 566, 24103, 227968, 655702, 655702, 227968, 24103, 566, 12, 1}

Mathematica

Clear[t, p, x, n]; p[x_, n_] = (1 + x)^(n + 1) + If[n < 2, 0, x*((1 - x)^(n + 1)*PolyLog[ -n, x]/x + (1 + x)^(n - 1))/2]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, -1, 10}]; Flatten[%]

Crossrefs

Sequence in context: A026637 A026659 A026386 * A283796 A156580 A157528

Adjacent sequences: A147529 A147530 A147531 * A147533 A147534 A147535

Keyword

nonn

Author

Roger L. Bagula, Nov 06 2008

Status

approved