A154816 A triangular sequence of polynomial coefficients: p(x,n)=((-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(5*m + 4)^n*x^m, {m, 0, Infinity}] + (-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(5*m + 1)^n*x^m, {m, 0, Infinity}]).

2, 5, 5, 17, 66, 17, 65, 685, 685, 65, 257, 6572, 16342, 6572, 257, 1025, 60675, 313300, 313300, 60675, 1025, 4097, 549418, 5340455, 10712060, 5340455, 549418, 4097, 16385, 4931825, 84856215, 303945575, 303945575, 84856215, 4931825, 16385

Offset

0,1

Comments

Row sums are:

{2, 10, 100, 1500, 30000, 750000, 22500000, 787500000, 31500000000,

1417500000000, 70875000000000}

Links

Table of n, a(n) for n=0..35.

Formula

p(x,n)=((-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(5*m + 4)^n*x^m, {m, 0, Infinity}] +

(-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(5*m + 1)^n*x^m, {m, 0, Infinity}]);

t(n,m)=coefficients(p(x,n))

Example

{2},
{5, 5},
{17, 66, 17},
{65, 685, 685, 65},
{257, 6572, 16342, 6572, 257},
{1025, 60675, 313300, 313300, 60675, 1025},
{4097, 549418, 5340455, 10712060, 5340455, 549418, 4097},
{16385, 4931825, 84856215, 303945575, 303945575, 84856215, 4931825, 16385},
{65537, 44136504, 1289970236, 7671841928, 13487971590, 7671841928, 1289970236, 44136504, 65537},
{262145, 394876735, 19055809150, 179073190690, 510225861280, 510225861280, 179073190690, 19055809150, 394876735, 262145},
{1048577, 3535716230, 276229994965, 3958390776760, 17297519264170, 27803646398596, 17297519264170, 3958390776760, 276229994965, 3535716230, 1048577}

Mathematica

Clear[p]; p[x_, n_] = ((-1)^( n + 1)*(x - 1)^(n + 1)*Sum[(5*m + 4)^n*x^m, {m, 0, Infinity}] +
(-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(5*m + 1)^n*x^m, {m, 0, Infinity}]);
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
Flatten[%]

Crossrefs

Sequence in context: A219586 A082534 A165659 * A367299 A140600 A056396

Adjacent sequences: A154813 A154814 A154815 * A154817 A154818 A154819

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Jan 15 2009

Status

approved