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A063421 Coefficient array for certain numerator polynomials N4(n,x), n >= 0 (rising powers of x) used for quadrinomials.

%I A063421 #13 May 09 2024 10:34:26
%S A063421 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 A063421 3,-4,1,10,-9,-15,27,-15,3,4,17,-60,66,-32,6,1,22,-41,-6,71,-74,36,-9,
%U A063421 1,15,6,-105,168,-111,24,9,-6,1,5,45,-147,133,21
%N A063421 Coefficient array for certain numerator polynomials N4(n,x), n >= 0 (rising powers of x) used for quadrinomials.
%C A063421 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 A063421 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 A063421 Row sums N4(n,1)=1 for all n.
%F A063421 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 A063421 The irregular triangle begins:
%e A063421   1;
%e A063421   1;
%e A063421   1;
%e A063421   1;
%e A063421   3, -3,  1;
%e A063421   2,  0, -2, 1;
%e A063421   1,  3, -5, 2;
%e A063421   6, -8,  3;
%e A063421   ...
%e A063421 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 A063421 Cf. A008287.
%K A063421 sign,easy,tabf
%O A063421 0,5
%A A063421 _Wolfdieter Lang_, Jul 27 2001