A063421 - OEIS

%I #13 May 09 2024 10:34:26

%S 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,

%T 3,-4,1,10,-9,-15,27,-15,3,4,17,-60,66,-32,6,1,22,-41,-6,71,-74,36,-9,

%U 1,15,6,-105,168,-111,24,9,-6,1,5,45,-147,133,21

%N Coefficient array for certain numerator polynomials N4(n,x), n >= 0 (rising powers of x) used for quadrinomials.

%C 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).

%C 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.

%C Row sums N4(n,1)=1 for all n.

%F 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).

%e The irregular triangle begins:

%e   1;

%e   1;

%e   1;

%e   1;

%e   3, -3,  1;

%e   2,  0, -2, 1;

%e   1,  3, -5, 2;

%e   6, -8,  3;

%e   ...

%e 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.

%Y Cf. A008287.

%K sign,easy,tabf

%O 0,5

%A _Wolfdieter Lang_, Jul 27 2001