OFFSET
0,4
COMMENTS
LINKS
Paul Drube, Generalized Path Pairs and Fuss-Catalan Triangles, arXiv:2007.01892 [math.CO], 2020. See Figure 4 p. 8 (up to signs).
FORMULA
Number triangle T(n, k) = (-1)^(n-k)*((3k+1)/(2n+k+1))*binomial(3n, n-k).
From Werner Schulte, Oct 27 2015: (Start)
If u(m,n) = (-1)^n*(Sum_{k=0..n} T(n,k)*((m+1)*k+1)) and v(m,n) = (-1)^n*(Sum_{k=0..n} (-1)^k*T(n,k)*m^k) and D(x) is the g.f. of A001764 then P(m,x) = Sum_{n>=0} u(m,n)*x^n = 1-(m+1)*x*D(x)^2 and Q(m,x) = Sum_{n>=0} v(m,n)*x^n = 1/P(m,x).
If G(k,x) is the g.f. of column k (k>=0) then G(k,x) = G(0,x)^(3*k+1). (End)
EXAMPLE
Triangle begins:
1;
-1, 1;
3, -4, 1;
-12, 18, -7, 1;
55, -88, 42, -10, 1;
-273, 455, -245, 75, -13, 1;
...
MAPLE
# Function RiordanSquare defined in A321620.
tt := sin(arcsin(3*sqrt(x*3/4))/3)/sqrt(x*3/4): R := RiordanSquare(tt, 11):
seq(seq(LinearAlgebra:-Row(R, n)[k]*(-1)^(n+k), k=1..n), n=1..11); # Peter Luschny, Nov 27 2018
MATHEMATICA
T[n_, k_] := (-1)^(n - k)((3k + 1)/(2n + k + 1)) Binomial[3n, n - k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)
PROG
(PARI) tabl(nn) = {my(m = matrix(nn, nn, n, k, if (n<k, 0, binomial(n+2*k-3, 3*k-3)))); m = 1/m; for (n=1, nn, for (k=1, n, print1(m[n, k], ", "); ); print(); ); } \\ Michel Marcus, Nov 20 2015
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Jul 06 2005
STATUS
approved