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A189231 Extended Catalan triangle read by rows.

%I A189231 #15 May 08 2020 17:30:01
%S A189231 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,
%T A189231 14,35,6,7,1,14,112,28,112,20,48,7,8,1,126,42,252,48,180,27,63,8,9,1,
%U A189231 42,420,90,480,75,270,35,80,9,10,1,462,132,990,165,825,110,385,44,99,10,11,1
%N A189231 Extended Catalan triangle read by rows.
%C A189231 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.
%C A189231 Then T(n,k) = S(n+1,k+1) for n>=0, k>=0.
%C A189231 The classical Catalan triangle A053121(n,k) can be recovered from this triangle by setting T(n,k) = 0 if n-k is odd.
%C A189231 The complementary Catalan triangle A189230(n,k) can be recovered from this triangle by setting T(n,k) = 0 if n-k is even.
%C A189231 T(n,0) are the extended Catalan numbers A057977(n).
%H A189231 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H A189231 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/TheLostCatalanNumbers">The lost Catalan numbers</a>
%F A189231 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).
%F A189231 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).
%F A189231 Row sums: A189911(n) = A162246(n,n) + A162246(n,n+1) for n>0.
%e A189231 The Laurent polynomials:
%e A189231 C(0,x) =                 0
%e A189231 C(1,x) =               x - 1/x
%e A189231 C(2,x) =         x^2 + x - 1/x - 1/x^2
%e A189231 C(3,x) = x^3 + 2 x^2 + x - 1/x - 2/x^2 -1/x^3
%e A189231 Triangle T(n,k) = S(n+1,k+1) starts
%e A189231 [0]   1,
%e A189231 [1]   1,  1,
%e A189231 [2]   1,  2,  1,
%e A189231 [3]   3,  2,  3,  1,
%e A189231 [4]   2,  8,  3,  4,  1,
%e A189231 [5]  10,  5, 15,  4,  5,  1,
%e A189231 [6]   5, 30,  9, 24,  5,  6,  1,
%e A189231 [7]  35, 14, 63, 14, 35,  6,  7, 1,
%e A189231     [0],[1],[2],[3],[4],[5],[6],[7]
%p A189231 A189231_poly := (n,x)-> `if`(n=0,0,(x+1/x)^(n-2)*(x-1/x)*(x+1/x+n-1)):
%p A189231 seq(print([n],seq(coeff(expand(A189231_poly(n,x)),x,k),k=1..n)),n=1..9);
%p A189231 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:
%p A189231 seq(print(seq(A189231(n,k),k=0..n)),n=0..9);
%t A189231 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 *)
%Y A189231 Cf. A053121, A162246, A057977, A189230.
%K A189231 nonn,tabl
%O A189231 0,5
%A A189231 _Peter Luschny_, May 01 2011