A213736 Triangle read by rows, coefficients of the Swiss-Knife median polynomials M_{n}(x) in descending order of powers.

1, 1, -1, -1, 1, -2, -5, 6, 4, 1, -3, -12, 29, 57, -72, -46, 1, -4, -22, 80, 261, -660, -1264, 1608, 1024, 1, -5, -35, 170, 775, -2941, -9385, 23880, 45620, -58080, -36976, 1, -6, -51, 310, 1815, -9186, -41033, 156618, 498660, -1269720, -2425056, 3087648

Offset: 0

Author

Peter Luschny, Jun 19 2012

Keywords

• sign
• tabf

Comments

M(n,0) = M(n,1) = A099023(n) = (-1)^n*A000657(n).

Examples

			M(0,x) = 1,
M(1,x) = x^2-x-1,
M(2,x) = x^4-2*x^3-5*x^2+6*x+4,
M(3,x) = x^6-3*x^5-12*x^4+29*x^3+57*x^2-72*x-46.
		

Links

• Peter Luschny, An old operation on sequences: the Seidel transform.

Crossrefs

Cf. A153641, A162660.

Programs

Maple

A213736_triangle := proc(n) local A, len, k, m, sk_poly;
len := 2*n-1; A := array(0..len,0..len);
sk_poly := proc(n, x) local v, k;
add(`if`((k+1)mod 4 = 0,0,(-1)^iquo(k+1,4))*2^iquo(-k,2)*
add((-1)^v*binomial(k,v)*(v+x+1)^n,v=0..k),k=0..n) end:
for m from 0 to len do A[m,0] := sk_poly(m,x);
   for k from m-1 by -1 to 0 do
       A[k,m-k] := A[k+1,m-k-1] - A[k,m-k-1] od od;
seq(print(seq(coeff(A[k,k],x,2*k-i),i=0..2*k)),k=0..n-1) end:
A213736_triangle(5);