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A064334 Triangle composed of generalized Catalan numbers.

%I A064334 #31 Sep 08 2022 08:45:04
%S A064334 1,1,1,1,1,1,1,0,1,1,1,1,-1,1,1,1,-2,5,-2,1,1,1,6,-25,13,-3,1,1,1,-18,
%T A064334 141,-98,25,-4,1,1,1,57,-849,826,-251,41,-5,1,1,1,-186,5349,-7448,
%U A064334 2817,-514,61,-6,1,1,1,622,-34825,70309,-33843,7206,-917,85,-7,1,1
%N A064334 Triangle composed of generalized Catalan numbers.
%C A064334 The sequence for column m (m >= 1) (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 (unphysical) 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. See also Liggett reference, proposition 3.19, p. 269, with lambda for alpha and rho for 1-beta.
%D A064334 T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Springer, 1999, p. 269.
%H A064334 G. C. Greubel, <a href="/A064334/b064334.txt">Rows n =0..100 of triangle, flattened</a>
%H A064334 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 A064334 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 A064334 G.f. for column m: (x^m)/(1-x*c(-m*x))= (x^m)*((m+1)+m*x*c(-m*x))/((m+1)-x), m>0, with the g.f. c(x) of Catalan numbers A000108.
%F A064334 T(n, m) = Sum_{k=0..n-m-1} (n-m-k)*binomial(n-m-1+k, k)*(-m)^k/(n-m), with T(n,0) = T(n,n)=1.
%F A064334 T(n,m) = (1/(1+m))^(n-m)*(1 + m*Sum_{k=0..n-m-1} C(k)*(-m*(m+1))^k ), n-m >= 1, T(n, n) = T(n,0) =1, T(n, m)=0 if n<m, with C(k)=A000108(k) (Catalan).
%F A064334 T(n, k) = hypergeometric([1-n+k, n-k], [-n+k], -k) if k<n else 1. - _Peter Luschny_, Nov 30 2014
%e A064334 Triangle starts:
%e A064334   1;
%e A064334   1,  1;
%e A064334   1,  1,  1;
%e A064334   1,  0,  1,  1;
%e A064334   1,  1, -1,  1, 1;
%e A064334   1, -2,  5, -2, 1, 1; ...
%t A064334 Table[If[k==0, 1, If[k==n, 1, Sum[(n-k-j)*Binomial[n-k-1+j, j]*(-k)^j/(n -k), {j, 0, n-k-1}]]], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, May 04 2019 *)
%o A064334 (Sage)
%o A064334 def T(n,k):
%o A064334     return hypergeometric([1-n, n], [-n], -k) if n>0 else 1
%o A064334 for n in (0..10):
%o A064334     print([simplify(T(n-k,k)) for k in (0..n)]) # _Peter Luschny_, Nov 30 2014
%o A064334 (PARI) {T(n,k) = if(k==0, 1, if(k==n, 1, sum(j=0, n-k-1, (n-k-j)* binomial(n-k-1+j, j)*(-k)^j/(n-k))))}; \\ _G. C. Greubel_, May 04 2019
%o A064334 (Magma) [[k eq 0 select 1 else k eq n select 1 else (&+[(n-k-j)* Binomial(n-k-1+j, j)*(-k)^j/(n-k): j in [0..n-k-1]]): k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, May 04 2019
%Y A064334 The unsigned column sequences (without leading zeros) are A000012, A064310-11, A064325-33 for m=0..11, respectively. Row sums (signed) give A064338. Row sums (unsigned) give A064339.
%Y A064334 Cf. A064062.
%K A064334 sign,easy,tabl
%O A064334 0,17
%A A064334 _Wolfdieter Lang_, Sep 21 2001