A174986 Numerator coefficients of an infinite sum polynomial:p(x,n)=Sum[(((1 + Sqrt[5])^k - (1 - Sqrt[5])^k)/(2^k* Sqrt[5]))^n*x^k, {k, 0, Infinity}]

-2, -1, 1, 5, -10, -5, -2, 8, 8, -2, -10, 70, 160, -70, -10, -1, 12, 53, -53, -12, 1, 20, -400, -3320, 6360, 3320, -400, -20, -2, 66, 984, -3568, -3568, 984, 66, -2, -10, 540, 14130, -92880, -178400, 92880, 14130, -540, -10, -4, 352, 15840, -184932, -654016

Offset

1,1

Comments

Row sums are:

{-2, 0, -10, 12, 140, 0, 5560, -5040, -150160, 0,...}.

The result seems to be a beta integer version of a Eulerian infinite sum.

Links

Table of n, a(n) for n=1..50.

Formula

p(x,n)=Sum[(((1 + Sqrt[5])^k - (1 - Sqrt[5])^k)/(2^k* Sqrt[5]))^n*x^k, {k, 0, Infinity}] out_n,m=Numerator_coefficients(p(x,n)/x)/2^(1 + Floor[n/2])

Example

{-2},
{-1, 1},
{5, -10, -5},
{-2, 8, 8, -2},
{-10, 70, 160, -70, -10},
{-1, 12, 53, -53, -12, 1},
{20, -400, -3320, 6360, 3320, -400, -20},
{-2, 66, 984, -3568, -3568, 984, 66, -2},
{-10, 540, 14130, -92880, -178400, 92880, 14130, -540, -10},
{-4, 352, 15840, -184932, -654016, 654016, 184932, -15840, -352, 4}

Mathematica

p[x_, n_] = Sum[(((1 + Sqrt[5])^k - (1 - Sqrt[5])^k)/(2^k*Sqrt[5]))^n*x^k, {k, 0, Infinity}];
Flatten[Table[CoefficientList[FullSimplify[Numerator[p[x, n]]/x], x]/2^(1 + Floor[n/2]), {n, 1, 10}]]

Crossrefs

Sequence in context: A098315 A006704 A324960 * A375321 A327671 A036563

Adjacent sequences: A174983 A174984 A174985 * A174987 A174988 A174989

Keyword

sign,tabl,uned

Author

Roger L. Bagula, Apr 03 2010

Status

approved