A063420 Coefficient array for certain numerator polynomials N3(n,x), n >= 0 (rising powers of x) used for trinomials.

1, 1, 1, 2, -1, 1, 1, -1, 3, -2, 1, 3, -4, 1, 4, -2, -2, 1, 1, 6, -9, 3, 5, 0, -9, 6, -1, 1, 10, -15, 3, 3, -1, 6, 5, -24, 18, -4, 1, 15, -20, -6, 18, -8, 1, 7, 14, -49, 36, -4, -4, 1, 1, 21, -21, -35, 60, -30, 5, 8, 28, -84, 50, 20, -30, 10, -1, 1, 28, -14

Offset: 0

Author

Wolfdieter Lang, Jul 27 2001

Keywords

• sign
• easy
• tabf

Comments

The g.f. of column k of array A027907(n,k) (trinomial coefficients) is (x^(ceiling(k/2)))*N3(k,x)/(1-x)^(k+1).

The sequence of degrees for the polynomials N3(n,x) is [0, 0, 0, 1, 2, 1, 3, 3, 3, 4, 5, 4, 6, 6, 6,...] for n >= 0.

Row sums N3(n,1)=1 for all n.

Examples

			{1}; {1}; {1}; {2,-1}; {1,1,-1}; {3,-2}; {1,3,-4,1}; {4,-2,-2,1}; ...
c=1: b(1,1)=1, b(1,2)=0.
N3(7,x)=4-2*x-2*x^2+5*x^3.
		

Formula

a(n, m) = [x^m]N3(n, x), n, m >= 0, with N3(n, x)= sum(((1-x)^(j-1))*(x^(b(c(n), j)))*N3(n-j, x), j=1..2), N3(n, x)= 1 for n=0, 1 and b(c(n), j) := 1 if 1<= j <= c(n) else 0, with c(n) := 1 if mod(n, 2)=0 else c(n) := mod(n, 2)-1; (hence b(0, j)=0, j=1..2).