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A064094
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Triangle composed of generalized Catalan numbers.
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24
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1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 14, 13, 4, 1, 1, 1, 42, 67, 25, 5, 1, 1, 1, 132, 381, 190, 41, 6, 1, 1, 1, 429, 2307, 1606, 413, 61, 7, 1, 1, 1, 1430, 14589, 14506, 4641, 766, 85, 8, 1, 1, 1, 4862, 95235
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OFFSET
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0,8
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COMMENTS
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The column m sequence (without leading zeros and the first 1) appears in the Derrida et al. 1992 reference as Z_{N}=Y_{N}(N+1), N >=0, for alpha = m, beta = 1 (or alpha = 1, beta = m). In the Derrida et al. 1993 reference the formula in eq. (39) gives Z_{N}(alpha,beta)/(alpha*beta)^N for N>=1.
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LINKS
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FORMULA
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G.f. for column m: (x^m)/(1-x*c(m*x))= (x^m)*((m-1)+m*x*c(m*x))/(m-1+x) with the g.f. c(x) of Catalan numbers A000108.
a(n, m)= sum((n-m-k)*binomial(n-m-1+k, k)*(m^k)/(n-m), k=0..n-m-1) = ((1/(1-m))^(n-m)*(1-m*sum(C(k)*(m*(1-m))^k, k=0..n-m-1)), n-m >= 1; a(n, n)=1; a(n, m)=0 if n<m; with C(k)=A000108(k) (Catalan).
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EXAMPLE
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Triangle begins:
1;
1, 1;
1, 1, 1;
1, 2, 1, 1;
1, 5, 3, 1, 1;
1, 14, 13, 4, 1, 1;
...
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MATHEMATICA
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a[n_, 0] = 1; a[n_, 1] := CatalanNumber[n - 1]; a[n_, n_] = 1; a[n_, m_] := (1/(1 - m))^(n - m)*(1 - m*Sum[ CatalanNumber[k]*(m*(1 - m))^k, {k, 0, n - m - 1}]); Table[ a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jul 05 2013 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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