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A226048
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Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (2,n)-rectangular grid with k '1's and (2n-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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41
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1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 6, 6, 6, 2, 1, 1, 2, 10, 14, 22, 14, 10, 2, 1, 1, 3, 15, 32, 60, 66, 60, 32, 15, 3, 1, 1, 3, 21, 55, 135, 198, 246, 198, 135, 55, 21, 3, 1, 1, 4, 28, 94, 266, 508, 777, 868, 777, 508, 266, 94, 28, 4, 1, 1, 4, 36
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OFFSET
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0,7
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COMMENTS
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Sum of rows (see example) gives A225826.
By columns:
Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 2 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Feb 16 2014
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LINKS
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FORMULA
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EXAMPLE
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n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0 1
1 1 1 1
2 1 1 3 1 1
3 1 2 6 6 6 2 1
4 1 2 10 14 22 14 10 2 1
5 1 3 15 32 60 66 60 32 15 3 1
6 1 3 21 55 135 198 246 198 135 55 21 3 1
7 1 4 28 94 266 508 777 868 777 508 266 94 28 4 1
8 1 4 36 140...
...
The length of row n is 2*n+1, so n>= floor((k+1)/2).
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MAPLE
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if type(k, 'even') then
binomial(2*n, k) +3*binomial(n, k/2) ;
else
binomial(2*n, k) +(1-(-1)^n)*binomial(n-1, (k-1)/2) ;
end if ;
%/4 ;
end proc:
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MATHEMATICA
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T[n_, k_] := If[EvenQ[k],
Binomial[2n, k] + 3 Binomial[n, k/2],
Binomial[2n, k] + (1-(-1)^n) Binomial[n-1, (k-1)/2]]/4;
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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