A154336 A triangular sequence of coefficients of polynomials: p(x,n)=(3*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k,0, Infinity}] -2*(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x).
1, 1, 1, 1, 10, 1, 1, 47, 47, 1, 1, 176, 558, 176, 1, 1, 597, 4442, 4442, 597, 1, 1, 1926, 29247, 65812, 29247, 1926, 1, 1, 6043, 173385, 747931, 747931, 173385, 6043, 1, 1, 18652, 965620, 7279396, 13712662, 7279396, 965620, 18652, 1, 1, 56993, 5173340, 64213532, 205619174, 205619174, 64213532, 5173340, 56993, 1
Offset: 0
Examples
{1},
{1, 1},
{1, 10, 1},
{1, 47, 47, 1},
{1, 176, 558, 176, 1},
{1, 597, 4442, 4442, 597, 1},
{1, 1926, 29247, 65812, 29247, 1926, 1},
{1, 6043, 173385, 747931, 747931, 173385, 6043, 1},
{1, 18652, 965620, 7279396, 13712662, 7279396, 965620, 18652, 1},
{1, 56993, 5173340, 64213532, 205619174, 205619174, 64213532, 5173340, 56993, 1}
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows
Programs
-
Mathematica
Clear[p, x, n]; p[x_, n_] = (3*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k, 0, Infinity}] - 2*(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n) * x^k, {k, 0,Infinity}]/x); Table[FullSimplify[ExpandAll[p[x, n]]], {n, 1, 10}]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}]; Flatten[%]
Formula
p(x,n)=(3*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k,0, Infinity}] -2*(x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x).
Functional form:
p(x,n)=(3*(-1)^n* 2^(-1 + n)* (-1 + x)^n* LerchPhi(x, 1 - n, 1/2) - 2*(-1)^(1 + n) *(-1 + x)^(1 + n)* PolyLog( -n, x)/x).
t(n,m)=Coefficients(p(x,n))
Comments