A226290 Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (3,n)-rectangular grid with k '1's and (3n-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, 2, 2, 1, 1, 2, 6, 6, 6, 2, 1, 1, 4, 13, 27, 39, 39, 27, 13, 4, 1, 1, 4, 22, 60, 139, 208, 252, 208, 139, 60, 22, 4, 1, 1, 6, 34, 129, 371, 794, 1310, 1675, 1675, 1310, 794, 371, 129, 34, 6, 1, 1, 6, 48, 218, 813, 2196, 4767, 8070, 11139, 12300, 11139, 8070, 4767, 2196, 813, 218, 48, 6, 1

Offset: 0

Author

Yosu Yurramendi, Jun 02 2013

Keywords

• nonn
• tabf

Comments

Sum of rows (see example) gives A225827.

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

By columns:

T(n,1) is A052928.

T(n,2) is A226292.

Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 3 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Apr 24 2015

Examples

			n\k 0 1  2   3   4    5    6    7     8     9    10   11   12
0   1
1   1 2  2   1
2   1 2  6   6   6    2    1
3   1 4 13  27  39   39   27   13     4     1
4   1 4 22  60 139  208  252  208   139    60    22    4    1
5   1 6 34 129 371  794 1310 1675  1675  1310   794  371  129    34   6   1
6   1 6 48 218 813 2196 4767 8070 11139 12300 11139 8070 4767  2196 813 218 48 6 1
...
The length of row n is 3*n+1.
		

Links

• Yosu Yurramendi, María Merino, Rows n = 0..30 of irregular triangle, flattened

Crossrefs

Cf. A225826, A225827, A005418, A034851, A226048.

Programs

Mathematica

T[n_, k_] := (Binomial[3n, k] + If[OddQ[n] || EvenQ[k], Binomial[Quotient[3 n, 2], Quotient[k, 2]], 0] + Sum[Binomial[n, k - 2i] Binomial[n, i] + Binomial[3 Mod[n, 2], k - 2i] Binomial[3 Quotient[n, 2], i], {i, 0, Quotient[k, 2]}])/4; Table[T[n, k], {n, 0, 6}, {k, 0, 3n}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)

PARI

T(n,k)={(binomial(3*n,k) + if(n%2==1||k%2==0,binomial(3*n\2,k\2),0) + sum(i=0,k\2, binomial(n,k-2*i) * binomial(1*n,i) + binomial(3*(n%2),k-2*i) * binomial(3*(n\2),i)))/4}
for(n=0,6,for(k=0,3*n, print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 30 2017

Extensions

Definition corrected by María Merino, May 19 2017