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A290789 A(n,k) is the n-th Carlitz-Riordan q-Catalan number (recurrence version) for q = -k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A290789 #31 Mar 14 2021 12:33:26
%S A290789 1,1,1,1,1,1,1,1,0,1,1,1,-1,-1,1,1,1,-2,-7,0,1,1,1,-3,-23,47,2,1,1,1,
%T A290789 -4,-55,586,873,0,1,1,1,-5,-109,3429,48778,-26433,-5,1,1,1,-6,-191,
%U A290789 13436,885137,-11759396,-1749159,0,1,1,1,-7,-307,40915,8425506,-904638963,-8596478231,220526159,14,1
%N A290789 A(n,k) is the n-th Carlitz-Riordan q-Catalan number (recurrence version) for q = -k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A290789 Alois P. Heinz, <a href="/A290789/b290789.txt">Antidiagonals n = 0..55, flattened</a>
%H A290789 J. Fürlinger, J. Hofbauer, <a href="http://dx.doi.org/10.1016/0097-3165(85)90089-5">q-Catalan numbers</a>, Journal of Combinatorial Theory, Series A, Volume 40, Issue 2, November 1985, Pages 248-264.
%H A290789 Robin Sulzgruber, <a href="http://othes.univie.ac.at/30616/">The Symmetry of the q,t-Catalan Numbers</a>, Masterarbeit, University of Vienna. Fakultät für Mathematik, 2013.
%F A290789 G.f. of column k: 1/(1-x/(1+k*x/(1-k^2*x/(1+k^3*x/(1-k^4*x/(1+ ... )))))).
%F A290789 A(n,k) = Sum_{j=0..n-1} A(j,k)*A(n-j-1,k)*(-k)^j for n>0, A(0,k) = 1.
%e A290789 Square array A(n,k) begins:
%e A290789   1,  1,   1,     1,      1,       1, ...
%e A290789   1,  1,   1,     1,      1,       1, ...
%e A290789   1,  0,  -1,    -2,     -3,      -4, ...
%e A290789   1, -1,  -7,   -23,    -55,    -109, ...
%e A290789   1,  0,  47,   586,   3429,   13436, ...
%e A290789   1,  2, 873, 48778, 885137, 8425506, ...
%p A290789 A:= proc(n, k) option remember; `if`(n=0, 1, add(
%p A290789       A(j, k)*A(n-j-1, k)*(-k)^j, j=0..n-1))
%p A290789     end:
%p A290789 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A290789 Unprotect[Power]; Power[0|0., 0|0.]=1; Protect[Power];A[n_, k_]:=A[n, k]=If[n==0 , 1, Sum[A[j, k] A[n - j - 1, k]* (-k)^j, {j, 0, n - 1}]]; Table[A[n, d - n], {d, 0, 15}, {n, 0, d}] (* _Indranil Ghosh_, Aug 13 2017 *)
%o A290789 (Python)
%o A290789 from sympy.core.cache import cacheit
%o A290789 @cacheit
%o A290789 def A(n, k):
%o A290789     return 1 if n==0 else sum(A(j, k)*A(n - j - 1, k)*(-k)**j for j in range(n))
%o A290789 for d in range(16): print([A(n, d - n) for n in range(d + 1)]) # _Indranil Ghosh_, Aug 13 2017
%Y A290789 Columns k=0-11 give: A000012, A090192, A015097, A015098, A015099, A015100, A015102, A015103, A015105, A015106, A015107, A015108.
%Y A290789 Main diagonal gives A290786.
%Y A290789 Cf. A290759.
%K A290789 sign,tabl
%O A290789 0,18
%A A290789 _Alois P. Heinz_, Aug 10 2017