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%I #31 Nov 30 2016 22:12:07
%S 0,0,0,0,0,0,0,2,2,0,1,14,39,14,1,3,66,208,208,66,3,12,198,794,1092,
%T 794,198,12,28,508,2196,3912,3912,2196,508,28,66,1092,5231,10626,
%U 13462,10626,5231,1092,66,126,2156,10808,24648,35787,35787,24648,10808,2156,126
%N Table T(n,k), n>=1, k>=1, read by antidiagonals: T(n,k) = number of equivalence classes of ways of placing five 1 X 1 tiles in an n X k rectangle under all symmetry operations of the rectangle.
%H Christopher Hunt Gribble, <a href="/A248017/b248017.txt">Table of n, a(n) for n = 1..9870</a>
%F Empirically,
%F T(n,k) = (4*k^5*n^5 - 40*k^4*n^4 + 140*k^3*n^3 + 2*k^5 + 20*k^4*n + 30*k^3*n^2 + 30*k^2*n^3 + 20*k*n^4 + 2*n^5 - 40*k^4 - 120*k^3*n - 185*k^2*n^2 - 120*k*n^3 - 40*n^4 + 160*k^3 - 20*k^2*n - 20*k*n^2 + 160*n^3 - 80*k^2 + 36*k*n - 80*n^2 + 48*k + 48*n + 45
%F + (- 30*k^2*n^3 - 20*k*n^4 - 2*n^5 - 15*k^2*n^2 + 120*k*n^3 + 40*n^4 + 20*k*n^2 - 160*n^3 + 60*k*n + 80*n^2 - 48*n - 45)*(-1)^k
%F + (- 2*k^5 - 20*k^4*n - 30*k^3*n^2 + 40*k^4 + 120*k^3*n - 15*k^2*n^2 - 160*k^3 + 20*k^2*n + 80*k^2 + 60*k*n - 48*k - 45)*(-1)^n
%F + (15*k^2*n^2 - 60*k*n + 45)*(-1)^k*(-1)^n)/1920;
%F T(1,k) = A005995(k-5) = (k-3)*(k-1)*((k-4)*(k-2)*2*k + 15*(1-(-1)^k))/480;
%F T(2,k) = A222715(k) = (k-2)*(k-1)*((2*k-3)(2*k-1)*2*k + 15*(1-(-1)^k))/120.
%e T(n,k) for 1<=n<=8 and 1<=k<=8 is:
%e . k 1 2 3 4 5 6 7 8 ...
%e n
%e 1 0 0 0 0 1 3 12 28
%e 2 0 0 2 14 66 198 508 1092
%e 3 0 2 39 208 794 2196 5231 10808
%e 4 0 14 208 1092 3912 10626 24648 50344
%e 5 1 66 794 3912 13462 35787 81648 164980
%e 6 3 198 2196 10626 35787 94248 212988 428076
%e 7 12 508 5231 24648 81648 212988 477903 955856
%e 8 28 1092 10808 50344 164980 428076 955856 1906128
%p b := proc (n::integer, k::integer)::integer;
%p (4*k^5*n^5 - 40*k^4*n^4 + 140*k^3*n^3 + 2*k^5 + 20*k^4*n
%p + 30*k^3*n^2 + 30*k^2*n^3 + 20*k*n^4 + 2*n^5 - 40*k^4
%p - 120*k^3*n - 185*k^2*n^2 - 120*k*n^3 - 40*n^4 + 160*k^3
%p - 20*k^2*n - 20*k*n^2 + 160*n^3 - 80*k^2 + 36*k*n - 80*n^2
%p + 48*k + 48*n + 45
%p + (- 30*k^2*n^3 - 20*k*n^4 - 2*n^5 - 15*k^2*n^2 + 120*k*n^3
%p + 40*n^4 + 20*k*n^2 - 160*n^3 + 60*k*n + 80*n^2 - 48*n
%p - 45)*(-1)^k
%p + (- 2*k^5 - 20*k^4*n - 30*k^3*n^2 + 40*k^4 + 120*k^3*n
%p - 15*k^2*n^2 - 160*k^3 + 20*k^2*n + 80*k^2 + 60*k*n
%p - 48*k - 45)*(-1)^n
%p + (15*k^2*n^2 - 60*k*n + 45)*(-1)^k*(-1)^n)/1920;
%p end proc;
%p seq(seq(b(n, k-n+1), n = 1 .. k), k = 1 .. 140);
%Y Cf. A034851, A226048, A226290, A225812, A228022, A228165, A228166, A243866, A006918, A244306, A244307, A248011, A248016, A248059, A248060, A248027.
%K tabl,nonn
%O 1,8
%A _Christopher Hunt Gribble_, Sep 30 2014
%E Terms corrected and extended by _Christopher Hunt Gribble_, Apr 16 2015
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