OFFSET
0,3
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
LINKS
Yosu Yurramendi, María Merino, Rows n = 0..30 of irregular triangle, flattened
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Yosu Yurramendi, Jun 02 2013
EXTENSIONS
Definition corrected by María Merino, May 19 2017
STATUS
approved