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

0,5

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]

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.

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)

Example

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;
  ...

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 *)

Prog

(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

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.

Sequence in context: A128198 A320031 A123349 * A114163 A189435 A279636

Adjacent sequences: A123349 A123350 A123351 * A123353 A123354 A123355

Keyword

nonn,tabl

Author

N. J. A. Sloane, Oct 14 2006

Extensions

More terms from Philippe Deléham, Apr 12 2007

Status

approved