A137277 Triangle of the coefficients [x^k] P_n(x) of the polynomials P_n(x) = 1/n * sum(j=0..floor(n/2), (-1)^j * binomial(n,j) * (n-4*j) * x^(n-2*j) ).

1, 0, 1, 2, 0, 1, 0, 1, 0, 1, -6, 0, 0, 0, 1, 0, -6, 0, -1, 0, 1, 20, 0, -5, 0, -2, 0, 1, 0, 25, 0, -3, 0, -3, 0, 1, -70, 0, 28, 0, 0, 0, -4, 0, 1, 0, -98, 0, 28, 0, 4, 0, -5, 0, 1, 252, 0, -126, 0, 24, 0, 9, 0, -6, 0, 1, 0, 378, 0, -150, 0, 15, 0, 15, 0, -7, 0, 1, -924, 0, 528, 0, -165, 0, 0, 0, 22, 0, -8, 0, 1, 0, -1452

Offset

0,4

Comments

The first four P_n(x) are the same as in A137276.

Row sums are 1, 1, 3, 2, -5, -6, 14, 20, -45, -70, 154, a signed variant of A047074.

Links

Table of n, a(n) for n=0..92.

Formula

P(0,n)=1. P_n(x) = 1/n*sum(j=0..floor(n/2), (-1)^j*binomial(n,j)*(n-4*j)*x^(n-2*j)).

Example

{1}, = 1
{0, 1}, = x
{2, 0, 1}, = 2+x^2
{0, 1, 0, 1}, = x+x^3
{-6, 0, 0, 0, 1}, = -6+x^4
{0, -6, 0, -1, 0, 1},
{20, 0, -5, 0, -2, 0, 1},
{0, 25, 0, -3,0, -3, 0, 1},
{-70, 0, 28, 0, 0, 0, -4, 0, 1},
{0, -98, 0, 28, 0,4, 0, -5, 0, 1},
{252, 0, -126, 0, 24, 0, 9, 0, -6, 0, 1}

Maple

A137277 := proc(n, k) if n = 0 then 1; else add( (-1)^j*binomial(n, j)*(n-4*j)*x^(n-2*j), j=0..n/2)/n ; coeftayl(%, x=0, k) ; fi; end:
seq( seq(A137277(n, k), k=0..n), n=0..15) ;

Mathematica

B[x_, n_] = If[n > 0, Sum[(-1)^p*Binomial[n, p]*(n - 4*p)*x^(n - 2*p)/ n, {p, 0, Floor[n/2]}], 1]; a = Table[CoefficientList[B[x, n], x], {n, 0, 10}]; Flatten[a]

Crossrefs

Cf. A138034.

Sequence in context: A287234 A309938 A140581 * A039975 A358679 A016253

Adjacent sequences: A137274 A137275 A137276 * A137278 A137279 A137280

Keyword

sign,easy,tabl

Author

Roger L. Bagula, Mar 13 2008

Extensions

Edited by the Associate Editors of the OEIS, Aug 27 2009

Status

approved