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A189231 Extended Catalan triangle read by rows. 8
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 (list; table; graph; refs; listen; history; text; internal format)
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).

REFERENCES

Peter Luschny, Divide, swing and conquer the factorial and the lcm{1,2,...,n}, preprint, April 2008.

LINKS

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

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-Fran├žois Alcover, Jul 30 2013 *)

CROSSREFS

Cf. A053121, A162246, A057977, A189230.

Sequence in context: A101608 A102853 A293390 * A107337 A066376 A151682

Adjacent sequences:  A189228 A189229 A189230 * A189232 A189233 A189234

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, May 01 2011

STATUS

approved

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Last modified February 18 17:08 EST 2018. Contains 299325 sequences. (Running on oeis4.)