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A290789
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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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14
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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
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OFFSET
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0,18
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LINKS
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J. Fürlinger, J. Hofbauer, q-Catalan numbers, Journal of Combinatorial Theory, Series A, Volume 40, Issue 2, November 1985, Pages 248-264.
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FORMULA
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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.
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EXAMPLE
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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, ...
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MAPLE
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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);
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MATHEMATICA
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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 *)
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PROG
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(Python)
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
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CROSSREFS
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Columns k=0-11 give: A000012, A090192, A015097, A015098, A015099, A015100, A015102, A015103, A015105, A015106, A015107, A015108.
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KEYWORD
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AUTHOR
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STATUS
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approved
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