A189231 Extended Catalan triangle read by rows.

1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 2, 8, 3, 4, 1, 10, 5, 15, 4, 5, 1, 5, 30, 9, 24, 5, 6, 1, 35, 14, 63, 14, 35, 6, 7, 1, 14, 112, 28, 112, 20, 48, 7, 8, 1, 126, 42, 252, 48, 180, 27, 63, 8, 9, 1, 42, 420, 90, 480, 75, 270, 35, 80, 9, 10, 1, 462, 132, 990, 165, 825, 110, 385, 44, 99, 10, 11, 1

Offset

0,5

Comments

Let S(n,k) denote the coefficients of the positive powers of the Laurent polynomials C_n(x) = (x+1/x)^(n-1)*(x-1/x)*(x+1/x+n) (if n>0) and C_0(x) = 0.

Then T(n,k) = S(n+1,k+1) for n>=0, k>=0.

The classical Catalan triangle A053121(n,k) can be recovered from this triangle by setting T(n,k) = 0 if n-k is odd.

The complementary Catalan triangle A189230(n,k) can be recovered from this triangle by setting T(n,k) = 0 if n-k is even.

T(n,0) are the extended Catalan numbers A057977(n).

Links

Table of n, a(n) for n=0..77.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Peter Luschny, The lost Catalan numbers

Formula

Recurrence: If k>n or k<0 then T(n,k) = 0 else if n=k then T(n,k) = 1; otherwise T(n,k) = T(n-1,k-1) + ((n-k) mod 2)*T(n-1,k) + T(n-1,k+1).

S(n,k) = (k/n)* A162246(n,k) for n>0 where S(n,k) are the coefficients from the definition provided the triangle A162246 is indexed in Laurent style by the recurrence: if abs(k) > n then A162246(n,k) = 0 else if n = k then A162246(n,k) = 1 and otherwise A162246(n,k) = A162246(n-1,k-1)+ modp(n-k,2) * A162246(n-1,k) + A162246(n-1,k+1).

Row sums: A189911(n) = A162246(n,n) + A162246(n,n+1) for n>0.

Example

The Laurent polynomials:
C(0,x) =                 0
C(1,x) =               x - 1/x
C(2,x) =         x^2 + x - 1/x - 1/x^2
C(3,x) = x^3 + 2 x^2 + x - 1/x - 2/x^2 -1/x^3
Triangle T(n,k) = S(n+1,k+1) starts
[0]   1,
[1]   1,  1,
[2]   1,  2,  1,
[3]   3,  2,  3,  1,
[4]   2,  8,  3,  4,  1,
[5]  10,  5, 15,  4,  5,  1,
[6]   5, 30,  9, 24,  5,  6,  1,
[7]  35, 14, 63, 14, 35,  6,  7, 1,
    [0],[1],[2],[3],[4],[5],[6],[7]

Maple

A189231_poly := (n, x)-> `if`(n=0, 0, (x+1/x)^(n-2)*(x-1/x)*(x+1/x+n-1)):
seq(print([n], seq(coeff(expand(A189231_poly(n, x)), x, k), k=1..n)), n=1..9);
A189231 := proc(n, k) option remember; `if`(k>n or k<0, 0, `if`(n=k, 1, A189231(n-1, k-1)+modp(n-k, 2)*A189231(n-1, k)+A189231(n-1, k+1))) end:
seq(print(seq(A189231(n, k), k=0..n)), n=0..9);

Mathematica

t[n_, k_] /; (k > n || k < 0) = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] = t[n-1, k-1] + Mod[n-k, 2]*t[n-1, k] + t[n-1, k+1]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2013 *)

Crossrefs

Cf. A053121, A162246, A057977, A189230.

Sequence in context: A304099 A293390 A363654 * A107337 A066376 A151682

Adjacent sequences: A189228 A189229 A189230 * A189232 A189233 A189234

Keyword

nonn,tabl

Author

Peter Luschny, May 01 2011

Status

approved