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%I #38 Oct 06 2017 04:28:01
%S 1,1,2,2,1,1,2,6,6,6,2,1,1,4,13,27,39,39,27,13,4,1,1,4,22,60,139,208,
%T 252,208,139,60,22,4,1,1,6,34,129,371,794,1310,1675,1675,1310,794,371,
%U 129,34,6,1,1,6,48,218,813,2196,4767,8070,11139,12300,11139,8070,4767,2196,813,218,48,6,1
%N Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (3,n)-rectangular grid with k '1's and (3n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.
%C Sum of rows (see example) gives A225827.
%C This triangle is to A225827 as Losanitsch's triangle A034851 is to A005418, and triangle A226048 to A225826.
%C By columns:
%C T(n,1) is A052928.
%C T(n,2) is A226292.
%C Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 3 rectangle under all symmetry operations of the rectangle. - _Christopher Hunt Gribble_, Apr 24 2015
%H Yosu Yurramendi, María Merino, <a href="/A226290/b226290.txt">Rows n = 0..30 of irregular triangle, flattened</a>
%e n\k 0 1 2 3 4 5 6 7 8 9 10 11 12
%e 0 1
%e 1 1 2 2 1
%e 2 1 2 6 6 6 2 1
%e 3 1 4 13 27 39 39 27 13 4 1
%e 4 1 4 22 60 139 208 252 208 139 60 22 4 1
%e 5 1 6 34 129 371 794 1310 1675 1675 1310 794 371 129 34 6 1
%e 6 1 6 48 218 813 2196 4767 8070 11139 12300 11139 8070 4767 2196 813 218 48 6 1
%e ...
%e The length of row n is 3*n+1.
%t T[n_, k_] := (Binomial[3n, k] + If[OddQ[n] || EvenQ[k], Binomial[Quotient[3 n, 2], Quotient[k, 2]], 0] + Sum[Binomial[n, k - 2i] Binomial[n, i] + Binomial[3 Mod[n, 2], k - 2i] Binomial[3 Quotient[n, 2], i], {i, 0, Quotient[k, 2]}])/4; Table[T[n, k], {n, 0, 6}, {k, 0, 3n}] // Flatten (* _Jean-François Alcover_, Oct 06 2017, after _Andrew Howroyd_ *)
%o (PARI)
%o T(n,k)={(binomial(3*n,k) + if(n%2==1||k%2==0,binomial(3*n\2,k\2),0) + sum(i=0,k\2, binomial(n,k-2*i) * binomial(1*n,i) + binomial(3*(n%2),k-2*i) * binomial(3*(n\2),i)))/4}
%o for(n=0,6,for(k=0,3*n, print1(T(n,k), ", "));print) \\ _Andrew Howroyd_, May 30 2017
%Y Cf. A225826, A225827, A005418, A034851, A226048.
%K nonn,tabf
%O 0,3
%A _Yosu Yurramendi_, Jun 02 2013
%E Definition corrected by _María Merino_, May 19 2017
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