A107985 Triangle read by rows: T(n,k) = (k+1)*(n+2)*(n-k+1)/2 for 0 <= k <= n.

1, 3, 3, 6, 8, 6, 10, 15, 15, 10, 15, 24, 27, 24, 15, 21, 35, 42, 42, 35, 21, 28, 48, 60, 64, 60, 48, 28, 36, 63, 81, 90, 90, 81, 63, 36, 45, 80, 105, 120, 125, 120, 105, 80, 45, 55, 99, 132, 154, 165, 165, 154, 132, 99, 55, 66, 120, 162, 192, 210, 216, 210, 192, 162, 120, 66

Offset: 0

Author

Emeric Deutsch, Jun 12 2005

Keywords

• nonn
• tabl

Comments

Kekulé numbers for certain benzenoids.

T(n,k) is the number of Dyck (n+3)-paths with 3 peaks (UDs) and last descent of length k+1. For example, T(1,1)=3 counts UUDUDUDD, UDUUDUDD, UDUDUUDD. The number of Dyck n-paths containing k peaks and with last descent of length j is (j/n)*binomial(n,k-1)*binomial(n-j-1,k-2) (where as usual binomial(a,b)=0 for b < 0 except that binomial(-1,-1):=1). - David Callan, Jun 26 2006

As a rectangular array, this is the accumulation array (cf. A144112) of the rectangular array W given by w(i,j)=i+j-1; i.e., W=A002024 as a rectangular array. - Clark Kimberling, Sep 16 2008

T(n,k) gives the dimension of an irreducible representation of SU(3) whose Young diagram (n,k) has two rows of length n and k, respectively. - Dimitris Cardaris, May 10 2025

Examples

			Triangle begins:
   1;
   3,  3;
   6,  8,  6;
  10, 15, 15, 10;
  15, 24, 27, 24, 15;
  ...
		

References

• S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 237, K{B(n,2,-l)}).

Links

• Volodymyr Mazorchuk and Xiaoyu Zhu, Combinatorics of infinite rank module categories over finite dimensional sl3-modules in Lie-algebraic context, arXiv:2501.00291 [math.RT], 2024. See page 9.

Crossrefs

Cf. A000217 (column 0 and main diagonal), A002024, A002415 (row sums), A098737, A144112.

Programs

Maple

T:=proc(n,k) if k<=n then (k+1)*(n+2)*(n-k+1)/2 else 0 fi end: for n from 0 to 11 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Mathematica

T[n_,k_]:= (k+1)(n+2)(n-k+1)/2; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten (* Stefano Spezia, Jan 06 2025 *)

Python

from math import isqrt, comb
def A107985(n):
    a = (m:=isqrt(k:=n+1<<1))+(k>m*(m+1))
    b = n-comb(a,2)
    return (b+1)*(a+1)*(a-b)>>1 # Chai Wah Wu, Jun 14 2025

Formula

T(n,n-k) = T(n,k); T(2n,n) = (n+1)^3.

G.f.: (1 - x^2*y)/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Oct 01 2023