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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.

%I A226048 #66 May 05 2023 09:43:25
%S A226048 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,
%T A226048 66,60,32,15,3,1,1,3,21,55,135,198,246,198,135,55,21,3,1,1,4,28,94,
%U A226048 266,508,777,868,777,508,266,94,28,4,1,1,4,36
%N 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.
%C A226048 Sum of rows (see example) gives A225826.
%C A226048 This triangle is to A225826 as Losanitsch's triangle A034851 is to A005418.
%C A226048 By columns:
%C A226048 T(n,1) is A004526.
%C A226048 T(n,2) is A000217.
%C A226048 T(n,3) is A225972.
%C A226048 T(n,4) is A071239.
%C A226048 T(n,5) is A222715.
%C A226048 T(n,6) is A228581.
%C A226048 T(n,7) is A228582.
%C A226048 T(n,8) is A228583.
%C A226048 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
%H A226048 Yosu Yurramendi and María Merino, <a href="/A226048/b226048.txt">Rows 0..40 of triangle, flattened</a>
%F A226048 If k even, 4*T(n,k) = binomial(2*n,k) +3*binomial(n,k/2). - _Yosu Yurramendi_, _María Merino_, Aug 25 2013
%F A226048 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]
%e A226048 n\k 0 1  2   3   4   5   6   7   8   9  10 11 12 13 14
%e A226048 0   1
%e A226048 1   1 1  1
%e A226048 2   1 1  3   1   1
%e A226048 3   1 2  6   6   6   2   1
%e A226048 4   1 2 10  14  22  14  10   2   1
%e A226048 5   1 3 15  32  60  66  60  32  15   3   1
%e A226048 6   1 3 21  55 135 198 246 198 135  55  21  3  1
%e A226048 7   1 4 28  94 266 508 777 868 777 508 266 94 28  4  1
%e A226048 8   1 4 36 140...
%e A226048 ...
%e A226048 The length of row n is 2*n+1, so n>= floor((k+1)/2).
%p A226048 A226048 := proc(n,k)
%p A226048     if type(k,'even') then
%p A226048         binomial(2*n,k) +3*binomial(n,k/2) ;
%p A226048     else
%p A226048         binomial(2*n,k) +(1-(-1)^n)*binomial(n-1,(k-1)/2) ;
%p A226048     end if ;
%p A226048     %/4 ;
%p A226048 end proc:
%p A226048 seq(seq(A226048(n,k),k=0..2*n),n=0..8) ; # _R. J. Mathar_, Jun 07 2020
%t A226048 T[n_, k_] := If[EvenQ[k],
%t A226048    Binomial[2n, k] + 3 Binomial[n, k/2],
%t A226048    Binomial[2n, k] + (1-(-1)^n) Binomial[n-1, (k-1)/2]]/4;
%t A226048 Table[T[n, k], {n, 0, 8}, { k, 0, 2n}] // Flatten (* _Jean-François Alcover_, May 05 2023 *)
%Y A226048 Cf. A225826, A005418, A034851.
%K A226048 nonn,tabf
%O A226048 0,7
%A A226048 _Yosu Yurramendi_, May 24 2013
%E A226048 Definition corrected by _María Merino_, May 19 2017