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 A226048 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. 41
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS Sum of rows (see example) gives A225826. This triangle is to A225826 as Losanitsch's triangle A034851 is to A005418. By columns: T(n,1) is A004526. T(n,2) is A000217. T(n,3) is A225972. T(n,4) is A071239. T(n,5) is A222715. T(n,6) is A228581. T(n,7) is A228582. T(n,8) is A228583. 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 LINKS Yosu Yurramendi and María Merino, Rows 0..40 of triangle, flattened FORMULA If k even, 4*T(n,k) = binomial(2*n,k) +3*binomial(n,k/2). - Yosu Yurramendi, María Merino, Aug 25 2013 If k odd, 4*T(n,k) = 4*T(n,k) = binomial(2*n,k) +(1-(-1)^n)*binomial(n-1,(k-1)/2). - Yosu Yurramendi, María Merino, Aug 25 2013 [corrected by Christian Barrientos, Jun 14 2018] EXAMPLE 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). MAPLE A226048 := proc(n, k)     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: seq(seq(A226048(n, k), k=0..2*n), n=0..8) ; # R. J. Mathar, Jun 07 2020 CROSSREFS Cf. A225826, A005418, A034851. Sequence in context: A026568 A138361 A030408 * A329210 A325669 A332488 Adjacent sequences:  A226045 A226046 A226047 * A226049 A226050 A226051 KEYWORD nonn,tabf AUTHOR Yosu Yurramendi, May 24 2013 EXTENSIONS Definition corrected by María Merino, May 19 2017 STATUS approved

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Last modified July 8 17:40 EDT 2020. Contains 335524 sequences. (Running on oeis4.)