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A063421
Coefficient array for certain numerator polynomials N4(n,x), n >= 0 (rising powers of x) used for quadrinomials.
5
1, 1, 1, 1, 3, -3, 1, 2, 0, -2, 1, 1, 3, -5, 2, 6, -8, 3, 3, 4, -16, 15, -6, 1, 1, 10, -20, 10, 3, -4, 1, 10, -9, -15, 27, -15, 3, 4, 17, -60, 66, -32, 6, 1, 22, -41, -6, 71, -74, 36, -9, 1, 15, 6, -105, 168, -111, 24, 9, -6, 1, 5, 45, -147, 133, 21
OFFSET
0,5
COMMENTS
The g.f. of column k of array A008287(n,k) (quadrinomial coefficients) is (x^(ceiling(k/3)))*N4(k,x)/(1-x)^(k+1).
The sequence of degrees for the polynomials N4(n,x) is [0, 0, 0, 0, 2, 3, 3, 2, 5, 6, 5, 5, 8, 8, 8,...] for n >= 0.
Row sums N4(n,1)=1 for all n.
FORMULA
a(n, m) = [x^m] N4(n, x), n, m >= 0, with N4(n, x) = Sum_{j=1..3} ((1-x)^(j-1))*(x^(b(c(n), j)))*N4(n-j, x), N4(n, x) = 1 for n = 0, 1, 2 and b(c(n), j) := 1 if 1<= j <= c(n) else 0, with c(n) := 2 if mod(n, 3) = 0 else c(n) := mod(n, 3) - 1; (hence b(0, j) = 0, j=1..3).
EXAMPLE
The irregular triangle begins:
1;
1;
1;
1;
3, -3, 1;
2, 0, -2, 1;
1, 3, -5, 2;
6, -8, 3;
...
For c=1: b(1,1) = 1, b(1,2) = 0 = b(1,3), and N4(6,x)=1+3*x-5*x^2+2*x^3.
CROSSREFS
Cf. A008287.
Sequence in context: A338114 A100940 A344390 * A368339 A244328 A073067
KEYWORD
sign,easy,tabf
AUTHOR
Wolfdieter Lang, Jul 27 2001
STATUS
approved