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%I #23 Aug 27 2019 06:06:42
%S 4,6,10,8,28,18,10,72,74,28,12,176,281,152,40,14,416,1020,762,270,54,
%T 16,960,3591,3664,1680,436,70,18,2176,12366,17120,10050,3238,658,88,
%U 20,4864,41877,78336,58500,23160,5677,944,108,22,10752,139968,352768,333750,161352,47236,9276,1302,130
%N Square array read by antidiagonals downwards: T(k,n) = sum of the site-perimeters of words of length n >= 1 over an alphabet of size k >= 1.
%H Andrew Howroyd, <a href="/A292767/b292767.txt">Table of n, a(n) for n = 1..1275</a>
%H Aubrey Blecher, Charlotte Brennan, Arnold Knopfmacher, and Toufik Mansour, <a href="http://dx.doi.org/10.22108/toc.2017.21465">The Site-Perimeter of Words</a>, Transactions on Combinatorics, Vol. 6 No. 2 (2017), pp. 37-48. ISSN (print): 2251-8657, ISSN (on-line): 2251-8665.
%F G.f. of row k: k*x*(36 + 12*k + (8 - 24*k - 8*k^2)*x + (2 - 5*k + 4*k^2 - k^3)*x^2)/(12*(1 - k*x)^2). - _Andrew Howroyd_, Oct 27 2018
%e Array begins (rows are indexed by k = 1,2,3,4,...; columns by n = 1,2,3,4,...):
%e 4, 6, 8, 10, 12, 14, 16, ...
%e 10, 28, 72, 176, 416, 960, 2176, ...
%e 18, 74, 281, 1020, 3591, 12366, 41877, ...
%e 28, 152, 762, 3664, 17120, 78336, 352768, ...
%e 40, 270, 1680, 10050, 58500, 333750, 1875000, ...
%e 54, 436, 3238, 23160, 161352, 1102464, 7420896, ...
%e 70, 658, 5677, 47236, 383131, 3049270, 23916361, ...
%e ...
%t RowGf[k_] := k x (36 + 12k + (8 - 24k - 8k^2) x + (2 - 5k + 4k^2 - k^3) x^2)/(12(1 - k x)^2);
%t T[k_, n_] := SeriesCoefficient[RowGf[k], {x, 0, n}];
%t Table[T[k - n + 1, n], {k, 1, 10}, {n, k, 1, -1}] // Flatten (* _Jean-François Alcover_, Aug 27 2019, from PARI *)
%o (PARI)
%o RowGf(k) = {k*x*(36 + 12*k + (8 - 24*k - 8*k^2)*x + (2 - 5*k + 4*k^2 - k^3)*x^2)/(12*(1 - k*x)^2)}
%o M(k,n)={Mat(vectorv(k,k,Vec(RowGf(k) + O(x*x^n))))}
%o { M(10,8) } \\ _Andrew Howroyd_, Oct 27 2018
%Y Row k=2 is A128135.
%K nonn,tabl
%O 1,1
%A _N. J. A. Sloane_, Sep 27 2017
%E Terms a(16) and beyond from _Andrew Howroyd_, Oct 27 2018