A063422 Coefficient array for certain numerator polynomials N5(n,x), n >= 0 (rising powers of x) used for quintinomials (also called pentanomials).

1, 1, 1, 1, 1, 4, -6, 4, -1, 3, -2, -2, 3, -1, 2, 2, -8, 7, -2, 1, 6, -14, 11, -3, 10, -20, 15, -4, 6, 2, -37, 65, -56, 28, -8, 1, 3, 16, -61, 78, -42, 0, 12, -6, 1, 1, 22, -57, 35, 42, -84, 60, -21, 3, 20, -25, -64, 196, -224, 136, -44, 6, 10, 35, -219, 420

Offset

0,6

Comments

The g.f. of column k of array A035343(n,k) (quintinomial coefficients) is (x^(ceiling(k/4)))*N5(k,x)/(1-x)^(k+1).

The sequence of degrees for the polynomials N5(n,x) is [0, 0, 0, 0, 0, 3, 4, 4, 4, 3, 7, 8, 8, 7, 7,...] for n >= 0.

Row sums N5(n,1)=1 for all n.

Links

Table of n, a(n) for n=0..65.

Formula

a(n, m) = [x^m]N5(n, x), n, m >= 0, with N5(n, x)= sum(((1-x)^(j-1))*(x^(b(c(n), j)))*N5(n-j, x), j=1..4), N5(n, x)= 1 for n=0..3 and b(c(n), j) := 1 if 1<= j <= c(n) else 0, with c(n) := 3 if mod(n, 4)=0 else c(n) := mod(n, 4)-1; (hence b(0, j)=0, j=1..4).

Example

{1}; {1}; {1}; {1}; {1}; {4,-6,4,-1}; {3,-2,-2,3,-1}; {2,2,-8,7,-2}; {1,6,-14,11,-3}; ...
c=2: b(2,1)= 1 = b(2,2), b(2,3)= 0 =b(2,4).
N5(6,x)=3-2*x-2*x^2+3*x^3-x^4.

Crossrefs

Sequence in context: A212635 A087108 A021687 * A261638 A010670 A240444

Adjacent sequences: A063419 A063420 A063421 * A063423 A063424 A063425

Keyword

sign,easy,tabf

Author

Wolfdieter Lang, Jul 27 2001

Status

approved