OFFSET
0,8
COMMENTS
This is the lower triangular Riordan matrix (f(t), t*f(t)), with f(t) = F^{[-1]}(t)/t, where F(x) = x/(1 - x^2 - x^3). The expansion of f(t) is given in A319201, the sequence of column k = 0.
This gives the inverse Matrix (with upper diagonals filled with 0's) of the Riordan matrix from A104578 for any finite dimension.
The A- and Z-sequences of this inverse Riordan triangle of (F(x)/x, F(x)) are A = [1, 0, -1, -1] generated by 1/(F(x)/x), and Z = [0,-1, -1] generated from 1/F(x) - 1/x. See the link W. Lang link for A- and Z- sequences in A006232 with references.
FORMULA
Recurrence from the Z- and A-sequence: T(n, k) = 0 if n < k; T(0, 0) = 1;
T(n, 0) = -(T(n-1, 1) + T(n-1, 2)), for n >= 1; and T(n, m) = T(n-1, k-1) - T(n-1, k+1) - T(n-1, k+2), for n>=1 and k >= 1.
Boas-Buck recurrence with B(n) = A319204(n): T(n, k) = ((k+1)/(n-k))*Sum_{j=k..n-1} b(n-1-j)*T(j, k), for n >= 1, k = 0,1, ..., n-1, and input T(n,n) = 1, for n >= 0.
G.f. of row polynomials R(n,x) = Sum_{k=0..n} T(n, k)*x^k is G(x,z) = f(z)/(1-x*z*f(z)) with the expansion of f given in A319201.
G.f. of column sequences Gcol(k, x) = x^k*f(x)^{k+1}, for k >= 0.
EXAMPLE
The triangle T(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 ...
-------------------------------------------------
0: 1
1: 0 1
2: -1 0 1
3: -1 -2 0 1
4: 2 -2 -3 0 1
5; 5 5 -3 -4 0 1
6: -2 12 9 -4 -5 0 1
7: -21 -7 21 14 -5 -6 0 1
8: -14 -56 -16 32 20 -6 -7 0 1
9: 72 -30 -108 -30 45 27 -7 -8 0 1
10: 138 210 -45 -180 -50 60 35 -8 -9 0 1
...
Recurrence from A- and Z-sequence: 5 = T(5, 0) = -(-2 + (-3)); 9 = T(6, 2) = 5 - (- 4 + 0).
Recurrence of Boas-Buck type, with B = [0,-2,-3, 6, ...] = A319204: 9 = T(6, 2) = ((2+1)/(6-2))*(6*1 + (-3)*0 + (-2)*(-3) + 0*(-3)) = (3/4)*12 = 9.
MATHEMATICA
(* The function RiordanArray is defined in A256893. *)
nmax = 10;
R = RiordanArray[1/(1 - #^2 - #^3)&, #/(1 - #^2 - #^3)&, nmax+1];
M = Inverse[PadRight[#, nmax+1]& /@ R];
T[n_, k_] := M[[n+1, k+1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 19 2019 *)
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Wolfdieter Lang, Oct 29 2018
STATUS
approved