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%I #36 Feb 24 2023 18:59:30
%S 1,1,1,1,2,1,1,3,5,1,1,4,14,14,1,1,5,30,84,42,1,1,6,55,330,594,132,1,
%T 1,7,91,1001,4719,4719,429,1,1,8,140,2548,26026,81796,40898,1430,1,1,
%U 9,204,5712,111384,884884,1643356,379236,4862,1
%N Triangle read by rows, giving Kekulé numbers for certain benzenoids (see the Cyvin-Gutman book for details).
%C There is another version in A078920. - _Philippe Deléham_, Apr 12 2007 [In other words, T(n,k) = A078920(n,n-k). - _Petros Hadjicostas_, Oct 19 2019]
%H G. C. Greubel, <a href="/A123352/b123352.txt">Rows n = 0..50 of the triangle, flattened</a>
%H Paul Barry, <a href="https://arxiv.org/abs/2011.10827">Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers</a>, arXiv:2011.10827 [math.CO], 2020.
%H S. J. Cyvin and I. Gutman, <a href="https://link.springer.com/book/10.1007/978-3-662-00892-8">Kekulé structures in benzenoid hydrocarbons</a>, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 183).
%H M. de Sainte-Catherine and G. Viennot, <a href="https://doi.org/10.1007/BFb0072509">Enumeration of certain Young tableaux with bounded height</a>, in: G. Labelle and P. Leroux (eds), <a href="https://doi.org/10.1007/BFb0072503">Combinatoire énumérative</a>, Lecture Notes in Mathematics, vol. 1234, Springer, Berlin, Heidelberg, 1986, pp. 58-67.
%F T(n, n-1) = A000108(n).
%F T(n, n-2) = A005700(n-1).
%F T(n, n-3) = A006149(n-2).
%F T(n, n-4) = A006150(n-3).
%F T(n, n-5) = A006151(n-4).
%F Triangle T(n,k) = (-1)^C(k+1,2) * Product{1 <= i <= j <= k} (-2*(n+1)+i+j)/(i+j). - _Paul Barry_, Jan 22 2009
%F From _G. C. Greubel_, Dec 17 2021: (Start)
%F T(n, k) = Product_{j=0..n-k-1} binomial(2*n-2*j, n-j)/binomial(n+j+1, n-j).
%F T(n, k) = ((n+1)!/(k+1)!)*Product_{j=0..n-k-1} Catalan(n-j)/binomial(n+j+1, n-j). (End)
%e Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 3, 5, 1;
%e 1, 4, 14, 14, 1;
%e 1, 5, 30, 84, 42, 1;
%e 1, 6, 55, 330, 594, 132, 1;
%e 1, 7, 91, 1001, 4719, 4719, 429, 1;
%e ...
%t A123352[n_, k_]:= Product[Binomial[2*n-2*j, n-j]/Binomial[n+j+1, n-j], {j, 0, n-k-1}];
%t Table[A123352[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 17 2021 *)
%o (Sage)
%o def A123352(n,k): return product( binomial(2*n-2*j, n-j)/binomial(n+j+1, n-j) for j in (0..n-k-1) )
%o flatten([[A123352(n,k) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Dec 17 2021
%Y Diagonals give A000108, A005700, A006149, A006150, A006151, etc.
%Y Columns include (truncated versions of) A000012 (k=0), A000027 (k=1), A000330 (k=2), A006858 (k=3), and A091962 (k=4).
%Y T(2n,n) gives A358597.
%Y Cf. A078920.
%K nonn,tabl
%O 0,5
%A _N. J. A. Sloane_, Oct 14 2006
%E More terms from _Philippe Deléham_, Apr 12 2007
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