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%I A064094 #21 Jan 07 2025 20:27:27
%S A064094 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,
%T A064094 381,190,41,6,1,1,1,429,2307,1606,413,61,7,1,1,1,1430,14589,14506,
%U A064094 4641,766,85,8,1,1,1,4862,95235,137089,55797,10746,1279,113,9,1,1
%N A064094 Triangle composed of generalized Catalan numbers.
%C A064094 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.
%H A064094 G. C. Greubel, <a href="/A064094/b064094.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A064094 B. Derrida, E. Domany, and D. Mukamel, <a href="http://dx.doi.org/10.1007/BF01050430">An exact solution of a one-dimensional asymmetric exclusion model with open boundaries</a>, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.
%H A064094 B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier, <a href="http://dx.doi.org/10.1088/0305-4470/26/7/011">Exact solution of a 1D asymmetric exclusion model using a matrix formulation</a>, J. Phys. A 26, 1993, 1493-1517; eq. (39), p. 1501, also appendix A1, (A12) p. 1513.
%F A064094 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.
%F A064094 T(n, m) = Sum_{j=0..n-m-1} (n-m-j)*binomial(n-m-1+j, j)*(m^j)/(n-m) or T(n, m) = (1/(1-m))^(n-m)*(1 - m*Sum_{j=0..n-m-1} C(j)*(m*(1-m))^j ), for n - m >= 1, T(n, n) = 1, T(n, m) = 0 if n<m; with C(k) = A000108(k) (Catalan).
%e A064094 Triangle begins:
%e A064094   1;
%e A064094   1,    1;
%e A064094   1,    1,     1;
%e A064094   1,    2,     1,     1;
%e A064094   1,    5,     3,     1,    1;
%e A064094   1,   14,    13,     4,    1,   1;
%e A064094   1,   42,    67,    25,    5,   1,   1;
%e A064094   1,  132,   381,   190,   41,   6,   1,   1;
%e A064094   1,  429,  2307,  1606,  413,  61,   7,   1,   1;
%e A064094   1, 1430, 14589, 14506, 4641, 766,  85,   8,   1,   1;
%t A064094 T[n_, 0] = 1; T[n_, 1] := CatalanNumber[n - 1]; T[n_, n_] = 1; T[n_, m_] := (1/(1 - m))^(n - m)*(1 - m*Sum[ CatalanNumber[k]*(m*(1 - m))^k, {k, 0, n - m - 1}]); Table[ T[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 05 2013 *)
%o A064094 (Magma)
%o A064094 function A064094(n,k)
%o A064094   if k eq 0 or k eq n then return 1;
%o A064094   else return (&+[(n-k-j)*Binomial(n-k-1+j, j)*k^j: j in [0..n-k-1]])/(n-k);
%o A064094   end if;
%o A064094 end function;
%o A064094 [A064094(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 27 2024
%o A064094 (SageMath)
%o A064094 def A064094(n,k):
%o A064094     if (k==0 or k==n): return 1
%o A064094     else: return sum((n-k-j)*binomial(n-k-1+j,j)*k^j for j in range(n-k))//(n-k)
%o A064094 flatten([[A064094(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Sep 27 2024
%Y A064094 Diagonals: A000012, A000012, A000027, A001844, A064096, A064302, A064303, A064304, A064305.
%Y A064094 Columns (without leading zeros): A000012 (k=0), A000108 (k=1), A064062 (k=2), A064063 (k=3), A064087 (k=4), A064088 (k=5), A064089 (k=6), A064090 (k=7), A064091 (k=8), A064092 (k=9), A064093 (k=10).
%Y A064094 Cf. A064095 (row sums).
%K A064094 nonn,easy,tabl
%O A064094 0,8
%A A064094 _Wolfdieter Lang_, Sep 13 2001

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