A154334 A triangular sequence of coefficients of polynomials: p(x,n) = ((x - 1)^n *(Sum_{k>=0} (((-1)^n*(2*k + 1)^(n - 1)))*x^k) + (x - 1)^(n + 1)*(Sum_{k>=0} ((-1)^(n + 1)*k^n)*x^k)/x)/2.

1, 1, 1, 1, 5, 1, 1, 17, 17, 1, 1, 51, 148, 51, 1, 1, 147, 992, 992, 147, 1, 1, 421, 5867, 12982, 5867, 421, 1, 1, 1213, 32475, 137671, 137671, 32475, 1213, 1, 1, 3527, 173110, 1286761, 2415602, 1286761, 173110, 3527, 1, 1, 10343, 902090, 11081582, 35361824, 35361824, 11081582, 902090, 10343, 1

Offset

0,5

Comments

Row sums are: {1, 2, 7, 36, 252, 2280, 25560, 342720, 5342400, 94711680, ...}.

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows

Formula

p(x,n) = ((x - 1)^n * (Sum_{k>=0} (((-1)^(n)*(2*k + 1)^(n - 1)))*x^k) + (x - 1)^(n + 1) * (Sum_{k>=0} ((-1)^(n + 1)*k^n)*x^k)/x)/2.

Functional form:

p(x,n) = ((-1)^n* 2^(-1 + n)* (-1 + x)^n* LerchPhi(x, 1 - n, 1/2) + (-1)^(1 + n) *(-1 + x)^(1 + n)* PolyLog(-n, x)/x)/2.

t(n,m) = Coefficients(p(x,n)).

Example

{1},
{1, 1},
{1, 5, 1},
{1, 17, 17, 1},
{1, 51, 148, 51, 1},
{1, 147, 992, 992, 147, 1},
{1, 421, 5867, 12982, 5867, 421, 1},
{1, 1213, 32475, 137671, 137671, 32475, 1213, 1},
{1, 3527, 173110, 1286761, 2415602, 1286761, 173110, 3527, 1},
{1, 10343, 902090, 11081582, 35361824, 35361824, 11081582, 902090, 10343, 1}

Mathematica

Clear[p, x, n]; p[x_, n_] = ((x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k, 0, Infinity}]
+ (x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x)/2;
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: A157181 A347974 A029847 * A178346 A168551 A262307

Adjacent sequences: A154331 A154332 A154333 * A154335 A154336 A154337

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 07 2009

Status

approved