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A094385 Triangle read by rows: T(n, k) = binomial(2*n, k-1)*binomial(2*n-k-1, n-k)/n for n, k >= 1, and T(n, 0) = 0^n.

%I A094385 #55 Sep 26 2024 15:35:05
%S A094385 1,0,1,0,1,2,0,2,6,5,0,5,20,28,14,0,14,70,135,120,42,0,42,252,616,770,
%T A094385 495,132,0,132,924,2730,4368,4004,2002,429,0,429,3432,11880,23100,
%U A094385 27300,19656,8008,1430,0,1430,12870,51051,116688,168300,157080,92820,31824,4862
%N A094385 Triangle read by rows: T(n, k) = binomial(2*n, k-1)*binomial(2*n-k-1, n-k)/n for n, k >= 1, and T(n, 0) = 0^n.
%H A094385 G. C. Greubel, <a href="/A094385/b094385.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A094385 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry2/barry231.html">A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays</a>, Journal of Integer Sequences, 16 (2013), #13.5.4.
%H A094385 Yue Cai and Catherine Yan, <a href="http://www.math.tamu.edu/~catherine.yan/Files/Borel-main-final.pdf">Counting with Borel's triangle</a>, Texas A&M University.
%H A094385 Yue Cai and Catherine Yan, <a href="https://arxiv.org/abs/1804.01597">Counting with Borel's triangle</a>, arXiv:1804.01597 [math.CO], 2018.
%H A094385 B. Derrida, E. Domany and D. Mukamel, <a href="http://dx.doi.org/10.1007/BF01050430">A 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 A094385 Jian Zhou, <a href="https://arxiv.org/abs/2108.10514">On Some Mathematics Related to the Interpolating Statistics</a>, arXiv:2108.10514 [math-ph], 2021.
%F A094385 T is given by [0, 1, 1, 1, 1, 1, 1, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is the operator defined in A084938.
%F A094385 Sum_{k = 0..n} T(n, k)*x^(n-k) = C(x+1; n), generalized Catalan numbers; see left diagonals of triangle A064094: A000012, A000108, A064062, A064063, A064087..A064093 for x = -1, 0, ..., 9, respectively.
%F A094385 From _G. C. Greubel_, Sep 26 2024: (Start)
%F A094385 T(n, 1) = A000108(n-1), n >= 1.
%F A094385 T(n, n-1) = A002694(n), n >= 1.
%F A094385 T(n, n) = A000108(n). (End)
%e A094385 Triangle begins:
%e A094385   1;
%e A094385   0,   1;
%e A094385   0,   1,    2;
%e A094385   0,   2,    6,     5;
%e A094385   0,   5,   20,    28,    14;
%e A094385   0,  14,   70,   135,   120,    42;
%e A094385   0,  42,  252,   616,   770,   495,   132;
%e A094385   0, 132,  924,  2730,  4368,  4004,  2002,  429;
%e A094385   0, 429, 3432, 11880, 23100, 27300, 19656, 8008, 1430; ...
%t A094385 T[n_, k_] := Binomial[2n, k-1] Binomial[2n-k-1, n-k]/n; T[0, 0] = 1;
%t A094385 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 19 2018 *)
%o A094385 (Magma)
%o A094385 A094385:= func< n,k | n eq 0 select 1 else Binomial(2*n, k-1)*Binomial(2*n-k-1, n-k)/n >;
%o A094385 [A094385(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 26 2024
%o A094385 (SageMath)
%o A094385 def A094385(n,k): return 1 if (n==0) else binomial(2*n,k-1)*binomial(2*n-k-1, n-k)//n
%o A094385 flatten([[A094385(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Sep 26 2024
%Y A094385 Variant of A062991, unsigned and transposed.
%Y A094385 See also A234950 for another version.
%Y A094385 Columns: A000007 (k=0), 2*A001700 (k=1).
%Y A094385 Diagonals: A002694 (k=n-1), A000108 (k=n).
%Y A094385 Row sums: A064062 (generalized Catalan C(2; n)).
%Y A094385 Cf. A000012, A002694, A064062, A064063, A064087, A064088, A064089.
%Y A094385 Cf. A064090, A064091, A064092, A064093, A064094.
%K A094385 easy,nonn,tabl
%O A094385 0,6
%A A094385 _Philippe Deléham_, Jun 03 2004, Jun 14 2007
%E A094385 New name using a formula of the author by _Peter Luschny_, Sep 26 2024