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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. 4

%I

%S 1,1,1,3,9,1,84,405,81,1,17550,121500,32805,729,1,25621596,247203171,

%T 82255257,2539107,6561,1,268715232324,3543210805275,1382411964132,

%U 53628242751,199290375,59049,1,21091830512086620,373203783345533355

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

%e Triangle T begins:

%e 1;

%e 1, 1;

%e 3, 9, 1;

%e 84, 405, 81, 1;

%e 17550, 121500, 32805, 729, 1;

%e 25621596, 247203171, 82255257, 2539107, 6561, 1;

%e 268715232324, 3543210805275, 1382411964132, 53628242751, 199290375, 59049, 1;

%e 21091830512086620, 373203783345533355, 165018275857291311, 7607829219099993, 36456526295226, 15884240049, 531441, 1; ...

%e where column 0 of T equals A179431(n) = C(3^(n-1), n):

%e [1, 1, 3, 84, 17550, 25621596, 268715232324, ...]. ...

%e Illustrate row n in column 0 of T^m equals C(m*3^(n-1), n) as follows.

%e Matrix square T^2 begins:

%e 1;

%e 2, 1;

%e 15, 18, 1;

%e 816, 1539, 162, 1;

%e 316251, 833490, 124659, 1458, 1;

%e 873642672, 3060203490, 585411786, 9861183, 13122, 1; ...

%e where column 0 of T^2 equals A179432(n) = C(2*3^(n-1), n):

%e [1, 2, 15, 816, 316251, 873642672, 17743125256857, ...]. ...

%e Matrix cube T^3 begins:

%e 1;

%e 3, 1;

%e 36, 27, 1;

%e 2925, 3402, 243, 1;

%e 1663740, 2667411, 275562, 2187, 1;

%e 6774333588, 14164214850, 1896890076, 21966228, 19683, 1; ...

%e where column 0 of T^3 equals A136393(n) = C(3^n, n):

%e [1, 3, 36, 2925, 1663740, 6774333588, 204208594169580, ...].

%o (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]}

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

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Jul 20 2010

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Last modified July 23 22:18 EDT 2019. Contains 325269 sequences. (Running on oeis4.)