A179430 Triangular matrix T where column 0 of T^m equals C(m*3^(n-1), n) at row n for n>=0, m>=0.

1, 1, 1, 3, 9, 1, 84, 405, 81, 1, 17550, 121500, 32805, 729, 1, 25621596, 247203171, 82255257, 2539107, 6561, 1, 268715232324, 3543210805275, 1382411964132, 53628242751, 199290375, 59049, 1, 21091830512086620, 373203783345533355

Offset: 0

Author

Paul D. Hanna, Jul 20 2010

Keywords

• nonn
• tabl

Examples

			Triangle T begins:
1;
1, 1;
3, 9, 1;
84, 405, 81, 1;
17550, 121500, 32805, 729, 1;
25621596, 247203171, 82255257, 2539107, 6561, 1;
268715232324, 3543210805275, 1382411964132, 53628242751, 199290375, 59049, 1;
21091830512086620, 373203783345533355, 165018275857291311, 7607829219099993, 36456526295226, 15884240049, 531441, 1; ...
where column 0 of T equals A179431(n) = C(3^(n-1), n):
[1, 1, 3, 84, 17550, 25621596, 268715232324, ...]. ...
Illustrate row n in column 0 of T^m equals C(m*3^(n-1), n) as follows.
Matrix square T^2 begins:
1;
2, 1;
15, 18, 1;
816, 1539, 162, 1;
316251, 833490, 124659, 1458, 1;
873642672, 3060203490, 585411786, 9861183, 13122, 1; ...
where column 0 of T^2 equals A179432(n) = C(2*3^(n-1), n):
[1, 2, 15, 816, 316251, 873642672, 17743125256857, ...]. ...
Matrix cube T^3 begins:
1;
3, 1;
36, 27, 1;
2925, 3402, 243, 1;
1663740, 2667411, 275562, 2187, 1;
6774333588, 14164214850, 1896890076, 21966228, 19683, 1; ...
where column 0 of T^3 equals A136393(n) = C(3^n, n):
[1, 3, 36, 2925, 1663740, 6774333588, 204208594169580, ...].
		

Crossrefs

Cf. A179431, A179432, A136393, A179433, A179434, variant: A136467.

Programs

PARI

{T(n, k)=local(M=matrix(n+1, n+1, r, c, binomial(r*3^(c-2), c-1)), P); P=matrix(n+1, n+1, r, c, binomial((r+1)*3^(c-2), c-1)); (P~*M~^-1)[n+1, k+1]}