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.

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

Offset: 0

Author

Yosu Yurramendi, May 24 2013

Keywords

• nonn
• tabf

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

Examples

			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).
		

Links

• Yosu Yurramendi and María Merino, Rows 0..40 of triangle, flattened

Crossrefs

Cf. A225826, A005418, A034851.

Programs

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

Mathematica

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;
Table[T[n, k], {n, 0, 8}, { k, 0, 2n}] // Flatten (* Jean-François Alcover, May 05 2023 *)

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]

Extensions

Definition corrected by María Merino, May 19 2017