A290786 -id:A290786 - cp's OEIS frontend

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.

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, -4, -55, 586, 873, 0, 1, 1, 1, -5, -109, 3429, 48778, -26433, -5, 1, 1, 1, -6, -191, 13436, 885137, -11759396, -1749159, 0, 1, 1, 1, -7, -307, 40915, 8425506, -904638963, -8596478231, 220526159, 14, 1

Offset: 0

Alois P. Heinz, Aug 10 2017

			Square array A(n,k) begins:
  1,  1,   1,     1,      1,       1, ...
  1,  1,   1,     1,      1,       1, ...
  1,  0,  -1,    -2,     -3,      -4, ...
  1, -1,  -7,   -23,    -55,    -109, ...
  1,  0,  47,   586,   3429,   13436, ...
  1,  2, 873, 48778, 885137, 8425506, ...

Alois P. Heinz, Antidiagonals n = 0..55, flattened
J. Fürlinger, J. Hofbauer, q-Catalan numbers, Journal of Combinatorial Theory, Series A, Volume 40, Issue 2, November 1985, Pages 248-264.
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Masterarbeit, University of Vienna. Fakultät für Mathematik, 2013.

Columns k=0-11 give: A000012, A090192, A015097, A015098, A015099, A015100, A015102, A015103, A015105, A015106, A015107, A015108.
Main diagonal gives A290786.
Cf. A290759.

A:= proc(n, k) option remember; `if`(n=0, 1, add(
      A(j, k)*A(n-j-1, k)*(-k)^j, j=0..n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

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 *)

from sympy.core.cache import cacheit
@cacheit
def A(n, k):
    return 1 if n==0 else sum(A(j, k)*A(n - j - 1, k)*(-k)**j for j in range(n))
for d in range(16): print([A(n, d - n) for n in range(d + 1)]) # Indranil Ghosh, Aug 13 2017

G.f. of column k: 1/(1-x/(1+k*x/(1-k^2*x/(1+k^3*x/(1-k^4*x/(1+ ... )))))).

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.

A290777 a(n) = n-th Carlitz-Riordan q-Catalan number (recurrence version) for q = n.

Original entry on oeis.org

1, 1, 3, 43, 5885, 12833546, 583552122727, 667480099386451779, 22507185898866512901924729, 25700910736350654917922270058287454, 1123582754598967452437582737448130799606015691, 2098715344599001562385695830901626594365732485934286582686

Offset: 0

Alois P. Heinz, Aug 10 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..36
J. Fürlinger, J. Hofbauer, q-Catalan numbers, Journal of Combinatorial Theory, Series A, Volume 40, Issue 2, November 1985, Pages 248-264.
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Masterarbeit, University of Vienna. Fakultät für Mathematik, 2013.

Main diagonal of A290759.

Cf. A090182, A290786.

b:= proc(n, k) option remember; `if`(n=0, 1, add(
      b(j, k)*b(n-j-1, k)*k^j, j=0..n-1))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..12);

b[n_, k_]:=b[n, k]=If[n==0, 1, Sum[b[j, k] b[n - j - 1, k]*k^j, {j, 0, n - 1}]]; Table[b[n, n], {n, 0, 15}] (* Indranil Ghosh, Aug 10 2017 *)

from sympy.core.cache import cacheit
@cacheit
def b(n, k):
    if n == 0:
        return 1
    return sum(b(j, k) * b(n - j - 1, k) * k**j for j in range(n))
def a(n): return b(n, n)
print([a(n) for n in range(16)]) # Indranil Ghosh, Aug 10 2017

a(n) = [x^n] 1/(1-x/(1-n*x/(1-n^2*x/(1-n^3*x/(1-n^4*x/(1- ... )))))).
a(n) = A290759(n,n) = A090182(2n,n).
a(n) ~ n^(n*(n-1)/2). - Vaclav Kotesovec, Aug 19 2017

cp's OEIS Frontend

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.

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