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A355881 Table read by descending antidiagonals: T(k,n) (k >= 0, n>= 1) is number of ways to (k+2)-color a 3 X n grid ignoring the variations of two colors.

%I A355881 #11 Sep 26 2022 20:54:43
%S A355881 1,1,2,1,9,3,1,41,49,4,1,187,801,169,5,1,853,13095,7141,441,6,1,3891,
%T A355881 214083,301741,38897,961,7,1,17749,3499929,12749989,3430789,153921,
%U A355881 1849,8,1,80963,57218481,538747549,302602093,24653151,488401,3249,9
%N A355881 Table read by descending antidiagonals: T(k,n) (k >= 0, n>= 1) is number of ways to (k+2)-color a 3 X n grid ignoring the variations of two colors.
%C A355881 With variations, the number of ways to color a 3 X 1 grid is (k+2)*(k+1)^2. The number of variations of two colors is (k+2)*(k+1). Therefore, T(k,1) = k+1. Only for k=1, the number of variations of two colors equals the number of permutations of all colors, see A020698.
%C A355881  T(0,n) = A000012(n) = constant 1
%C A355881 T(1,n) = A020698(n-1)
%C A355881 T(2,n) = A355882(n)
%C A355881 T(3,n) = A355883(n)
%H A355881 Gerhard Kirchner, <a href="/A355881/a355881.pdf">Derivation of the recurrence</a>
%F A355881 T(k,n) = k*(k^2 + k + 3) * T(k,n-1) - (k^4 + k^3 + k^2-1) * T(k,n-2)
%F A355881 with T(k,1) = k+1, T(k,2) = (k^2+k+1)^2.
%F A355881 G.f.: x*(k + 1 - (k^2 + k - 1)*x) / (1 - k*(k^2 + k + 3)*x + (k^4 + k^3 + k^2 - 1)*x^2).
%e A355881 Table begins:
%e A355881 k\n_1____2______3_________4___________5_____________6________________7
%e A355881 0:  1    1      1         1           1             1                1
%e A355881 1:  2    9     41       187         853          3891            17749
%e A355881 2:  3   49    801     13095      214083       3499929         57218481
%e A355881 3:  4  169   7141    301741    12749989     538747549      22764640981
%e A355881 4:  5  441  38897   3430789   302602093   26690078241    2354115497017
%e A355881 5:  6  961 153921  24653151  3948635061  632443246191  101296892084301
%e A355881 6:  7 1849 488401 129007867 34076567743 9001098120361 2377580042199049
%Y A355881 Cf. A000012, A020698, A355882, A355883.
%K A355881 nonn,tabl
%O A355881 0,3
%A A355881 _Gerhard Kirchner_, Jul 24 2022