A123352 Triangle read by rows, giving Kekulé numbers for certain benzenoids (see the Cyvin-Gutman book for details).

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, 1, 7, 91, 1001, 4719, 4719, 429, 1, 1, 8, 140, 2548, 26026, 81796, 40898, 1430, 1, 1, 9, 204, 5712, 111384, 884884, 1643356, 379236, 4862, 1

Offset: 0

Author

N. J. A. Sloane, Oct 14 2006

Keywords

• nonn
• tabl

Comments

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]

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
  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;
  1, 7, 91, 1001, 4719, 4719, 429, 1;
  ...
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• Paul Barry, Notes on the Hankel transform of linear combinations of consecutive pairs of Catalan numbers, arXiv:2011.10827 [math.CO], 2020.
• S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 183).
• M. de Sainte-Catherine and G. Viennot, Enumeration of certain Young tableaux with bounded height, in: G. Labelle and P. Leroux (eds), Combinatoire énumérative, Lecture Notes in Mathematics, vol. 1234, Springer, Berlin, Heidelberg, 1986, pp. 58-67.

Crossrefs

Diagonals give A000108, A005700, A006149, A006150, A006151, etc.

Columns include (truncated versions of) A000012 (k=0), A000027 (k=1), A000330 (k=2), A006858 (k=3), and A091962 (k=4).

T(2n,n) gives A358597.

Cf. A078920.

Programs

Mathematica

A123352[n_, k_]:= Product[Binomial[2*n-2*j, n-j]/Binomial[n+j+1, n-j], {j, 0, n-k-1}];
Table[A123352[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 17 2021 *)

Sage

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) )
flatten([[A123352(n,k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Dec 17 2021

Formula

T(n, n-1) = A000108(n).

T(n, n-2) = A005700(n-1).

T(n, n-3) = A006149(n-2).

T(n, n-4) = A006150(n-3).

T(n, n-5) = A006151(n-4).

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

From G. C. Greubel, Dec 17 2021: (Start)

T(n, k) = Product_{j=0..n-k-1} binomial(2*n-2*j, n-j)/binomial(n+j+1, n-j).

T(n, k) = ((n+1)!/(k+1)!)*Product_{j=0..n-k-1} Catalan(n-j)/binomial(n+j+1, n-j). (End)

Extensions

More terms from Philippe Deléham, Apr 12 2007