A109956 Inverse of Riordan array (1/(1-x), x/(1-x)^3), A109955.

1, -1, 1, 3, -4, 1, -12, 18, -7, 1, 55, -88, 42, -10, 1, -273, 455, -245, 75, -13, 1, 1428, -2448, 1428, -510, 117, -16, 1, -7752, 13566, -8379, 3325, -910, 168, -19, 1, 43263, -76912, 49588, -21252, 6578, -1472, 228, -22, 1, -246675, 444015, -296010, 134550, -45630, 11700, -2223, 297, -25, 1

Offset: 0

Author

Paul Barry, Jul 06 2005

Keywords

• easy
• sign
• tabl

Comments

Riordan array (g,f) where f/(1-f)^3=x and g=1-f.

First column is (-1)^n*binomial(3n,n)/(2n+1), a signed version of A001764.

Second column is a signed version of A006629.

Diagonal sums are A109957.

Examples

			Triangle begins:
      1;
     -1,   1;
      3,  -4,    1;
    -12,  18,   -7,   1;
     55, -88,   42, -10,   1;
   -273, 455, -245,  75, -13, 1;
   ...
		

Links

• Paul Barry, d-orthogonal polynomials, Fuss-Catalan matrices and lattice paths, arXiv:2505.16718 [math.CO], 2025. See p. 13.
• Paul Drube, Generalized Path Pairs and Fuss-Catalan Triangles, arXiv:2007.01892 [math.CO], 2020. See Figure 4 p. 8 (up to signs).

Crossrefs

Cf. A001764, A109955, A109957, A321620.

Programs

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 *)

PARI

tabl(nn) = {my(m = matrix(nn, nn, n, k, if (nMichel Marcus, Nov 20 2015

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)