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%I #23 May 27 2019 02:06:34
%S 1,1,1,2,3,3,1,4,7,9,3,1,10,23,29,16,5,1,25,69,99,62,27,5,1,70,229,
%T 351,275,132,39,7,1,196,731,1249,1121,643,221,55,7,1,574,2385,4437,
%U 4584,2997,1278,367,72,9,1,1681,7657,15597,18012,13458,6678,2322,540,93,9,1
%N Irregular triangle read by rows giving the total number of isomers (nonisomorphic systems) of unbranched k-4-catafusenes.
%C From _Petros Hadjicostas_, May 26 2019: (Start)
%C Let I(r, k) be the total number of isomers (nonisomorphic systems) of unbranched k-4-catafusenes, which are generated from catafusenes by converting k of its r hexagons to tetragons. According to Cyvin et al. (1996), for r >= k, we have I(r, k) = 1/4 *(binomial(r, k) + (r - 2)! * (r^2 + (4 * k - 1) * r + 4 * k * (k - 2)) * 3^(r - k - 2)/(k! * (r - k)!) + (2 + (-1)^k - (-1)^r) * (binomial(floor(r/2), floor(k/2)) + 2 * binomial(floor(r/2) - 1, floor(k/2) - 1)) * 3^(floor(r/2) - floor(k/2) - 1)). See Eq. (48) on p. 503 in the paper.
%C Letting k = 0 - 10, we get the eleven columns of Table 2 on p. 501 of Cyvin et al. (1996). (We need r >= max(k, 2) because the number of hexagons r should be greater than or equal to the number of converted polygons k.)
%C (End)
%H S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, <a href="https://dx.doi.org/10.1016/0022-2860(95)09039-8">Isomer enumeration of some polygonal systems representing polycyclic conjugated hydrocarbons</a>, Journal of Molecular structure 376 (Issues 1-3) (1996), 495-505. See Table 2 on p. 501.
%F For the element T(n, k) in row n >= 2 and column k >= 0 (such that max(k, 2) <= n), we have T(n, k) = I(r = n, k), where I(r, k) is given above in the comments. - _Petros Hadjicostas_, May 26 2019
%e Triangle begins (rows start at n = 2 and columns at k = 0):
%e 1, 1, 1;
%e 2, 3, 3, 1;
%e 4, 7, 9, 3, 1;
%e 10, 23, 29, 16, 5, 1;
%e 25, 69, 99, 62, 27, 5, 1;
%e 70, 229, 351, 275, 132, 39, 7, 1;
%e 196, 731, 1249, 1121, 643, 221, 55, 7, 1;
%e 574, 2385, 4437, 4584, 2997, 1278, 367, 72, 9, 1;
%e 1681, 7657, 15597, 18012, 13458, 6678, 2322, 540, 93, 9, 1;
%e ...
%Y Column k = 0 is A001998. Column k = 3 is A323941.
%K nonn,tabf
%O 2,4
%A _N. J. A. Sloane_, Feb 09 2019
%E Name edited by _Petros Hadjicostas_, May 26 2019
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