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

%I A107985 #45 Jun 14 2025 17:15:43
%S A107985 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,
%T A107985 60,48,28,36,63,81,90,90,81,63,36,45,80,105,120,125,120,105,80,45,55,
%U A107985 99,132,154,165,165,154,132,99,55,66,120,162,192,210,216,210,192,162,120,66
%N A107985 Triangle read by rows: T(n,k) = (k+1)*(n+2)*(n-k+1)/2 for 0 <= k <= n.
%C A107985 Kekulé numbers for certain benzenoids.
%C A107985 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
%C A107985 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
%C A107985 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
%D A107985 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)}).
%H A107985 Volodymyr Mazorchuk and Xiaoyu Zhu, <a href="https://arxiv.org/abs/2501.00291">Combinatorics of infinite rank module categories over finite dimensional sl3-modules in Lie-algebraic context</a>, arXiv:2501.00291 [math.RT], 2024. See page 9.
%F A107985 T(n,n-k) = T(n,k); T(2n,n) = (n+1)^3.
%F A107985 G.f.: (1 - x^2*y)/((1 - x)^3*(1 - x*y)^3). - _Stefano Spezia_, Oct 01 2023
%e A107985 Triangle begins:
%e A107985    1;
%e A107985    3,  3;
%e A107985    6,  8,  6;
%e A107985   10, 15, 15, 10;
%e A107985   15, 24, 27, 24, 15;
%e A107985   ...
%p A107985 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
%t A107985 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 *)
%o A107985 (Python)
%o A107985 from math import isqrt, comb
%o A107985 def A107985(n):
%o A107985     a = (m:=isqrt(k:=n+1<<1))+(k>m*(m+1))
%o A107985     b = n-comb(a,2)
%o A107985     return (b+1)*(a+1)*(a-b)>>1 # _Chai Wah Wu_, Jun 14 2025
%Y A107985 Cf. A000217 (column 0 and main diagonal), A002024, A002415 (row sums), A098737, A144112.
%K A107985 nonn,tabl
%O A107985 0,2
%A A107985 _Emeric Deutsch_, Jun 12 2005