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A290777
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a(n) = n-th Carlitz-Riordan q-Catalan number (recurrence version) for q = n.
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4
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1, 1, 3, 43, 5885, 12833546, 583552122727, 667480099386451779, 22507185898866512901924729, 25700910736350654917922270058287454, 1123582754598967452437582737448130799606015691, 2098715344599001562385695830901626594365732485934286582686
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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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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a(n) = [x^n] 1/(1-x/(1-n*x/(1-n^2*x/(1-n^3*x/(1-n^4*x/(1- ... )))))).
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MAPLE
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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);
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MATHEMATICA
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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 *)
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PROG
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(Python)
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)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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