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Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (4,n)-rectangular grid with k '1's and (4n-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.
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%I #42 Oct 06 2017 04:28:10

%S 1,1,2,4,2,1,1,2,10,14,22,14,10,2,1,1,4,22,60,139,208,252,208,139,60,

%T 22,4,1,1,4,36,140,476,1092,2044,2860,3270,2860,2044,1092,476,140,36,

%U 4,1,1,6,56,294,1253,3912,9808,19464,31706,42116,46448,42116,31706

%N Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (4,n)-rectangular grid with k '1's and (4n-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 The length of row n is 4*n+1.

%C Sum of rows (see example) gives A225828.

%C This triangle is to A225828 as Losanitsch's triangle A034851 is to A005418, triangle A226048 to A225826, and triangle A226290 to A225827.

%C Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 4 rectangle under all symmetry operations of the rectangle. - _Christopher Hunt Gribble_, Apr 24 2015

%H Yosu Yurramendi and María Merino, <a href="/A225812/b225812.txt">Rows n = 0..26 of irregular triangle, flattened</a>

%e Irregular triangle:

%e 1

%e 1 2 4 2 1

%e 1 2 10 14 22 14 10 2 1

%e 1 4 22 60 139 208 252 208 139 60 22 4 1

%e 1 4 36 140 476 1092 2044 2860 3270 2860 2044 1092 476 140 36 4 1 ...

%t T[n_, k_] := (Binomial[4n, k] + If[EvenQ[k], 2 Binomial[2n, k/2], 0] + Sum[Binomial[4 Mod[n, 2], k - 2i] Binomial[4 Quotient[n, 2], i], {i, 0, Quotient[k, 2]}])/4; Table[T[n, k], {n, 0, 5}, {k, 0, 4n}] // Flatten (* _Jean-François Alcover_, Oct 06 2017, after _Andrew Howroyd_ *)

%o (PARI)

%o T(n,k)={(binomial(4*n,k) + if(k%2==0,2*binomial(2*n,k/2),0) + sum(i=0,k\2,binomial(4*(n%2),k-2*i)*binomial(4*(n\2),i)))/4}

%o for(n=0,4,for(k=0,4*n, print1(T(n,k), ", "));print) \\ _Andrew Howroyd_, May 30 2017

%Y Cf. A225826, A225827, A225828, A005418, A034851, A226048.

%K nonn,tabf

%O 0,3

%A _Yosu Yurramendi_, _María Merino_, Jul 30 2013

%E Definition corrected by _María Merino_, May 19 2017