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Triangle read by rows: T(n,k) = (k+1)(n+2)(n-k+1)/2 for 0 <= k <= n.
9

%I #31 Oct 01 2023 09:23:49

%S 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 60,48,28,36,63,81,90,90,81,63,36,45,80,105,120,125,120,105,80,45,55,

%U 99,132,154,165,165,154,132,99,55,66,120,162,192,210,216,210,192,162,120,66

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

%C Kekulé numbers for certain benzenoids.

%C 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 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

%D 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)}).

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

%F G.f.: (1 - x^2*y)/((1 - x)^3*(1 - x*y)^3). - _Stefano Spezia_, Oct 01 2023

%e Triangle begins:

%e 1;

%e 3, 3;

%e 6, 8, 6;

%e 10, 15, 15, 10;

%e 15, 24, 27, 24, 15;

%e ...

%p 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

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

%K nonn,tabl

%O 0,2

%A _Emeric Deutsch_, Jun 12 2005