A274480 Triangle T, read by rows, where T(n,k) = [T^(3^k)](n-k,0) * (3^k)^(n-k) for n>=k>=0 such that row n of the 3^(n-1)-th root of T consists solely of integers given by: [T^( 1/3^(n-1) )](n,k) = (3^k)^(n-k) for n>=0.

1, 1, 1, 6, 9, 1, 201, 405, 81, 1, 43668, 108135, 30618, 729, 1, 66109077, 192068901, 69343209, 2421009, 6561, 1, 734489285949, 2429869742037, 1055300462694, 48233053719, 194507406, 59049, 1, 62046990518394987, 228954896130792105, 115264903237128999, 6477074077667103, 34597553648841, 15712053165, 531441, 1, 40856017343540753635650, 165659766162266374832070, 94247154749939415534567, 6256382300132639786847, 41519988501386251608, 25084397696688135, 1271514044898, 4782969, 1

Offset

0,4

Links

Paul D. Hanna, Table of n, a(n) for n = 0..495, for rows 0..30 of the flattened triangle.

Formula

The value of (3^m)-th matrix power of T at row n and column k is related to row n+m and column k+m of T by: [T^(3^m)](n,k) = T(n+m,k+m)/(3^m)^(n-k) for m>=0.

Example

Below we illustrate this triangle and its 2 main properties:
(1) [T^(3^m)](n,k) = T(n+m,k+m)/(3^m)^(n-k) for m>=0;
(2) [T^( 1/3^(n-1) )](n,k) = (3^k)^(n-k) for n>=k>=0.
This triangle begins:
1;
1, 1;
6, 9, 1;
201, 405, 81, 1;
43668, 108135, 30618, 729, 1;
66109077, 192068901, 69343209, 2421009, 6561, 1;
734489285949, 2429869742037, 1055300462694, 48233053719, 194507406, 59049, 1;
62046990518394987, 228954896130792105, 115264903237128999, 6477074077667103, 34597553648841, 15712053165, 531441, 1;
40856017343540753635650, 165659766162266374832070, 94247154749939415534567, 6256382300132639786847, 41519988501386251608, 25084397696688135, 1271514044898, 4782969, 1;
...
(1) Illustrate [T^(3^m)](n,k) = T(n+m,k+m)/(3^m)^(n-k) as follows.
Matrix cube, T^3, begins:
1;
3, 1;
45, 27, 1;
4005, 3402, 243, 1;
2371221, 2568267, 269001, 2187, 1;
9999463959, 13028400774, 1786409397, 21611934, 19683, 1; ...
where [T^(3^1)](n,k) = T(n+1,k+1)/3^(n-k).
Matrix 9th power, T^9, begins:
1,
9, 1,
378, 81, 1,
95121, 29889, 729, 1,
160844454, 66163311, 2401326, 6561, 1,
1952021257551, 987208364223, 47458921329, 193975965, 59049, 1; ...
where [T^(3^2)](n,k) = T(n+2,k+2)/9^(n-k).
Matrix 27th power, T^27, begins:
1,
27, 1,
3321, 243, 1,
2450493, 266814, 2187, 1,
12187757583, 1757737827, 21552885, 19683, 1,
436018039571421, 78127183452888, 1274419432845, 1744189362, 177147, 1; ...
where [T^(3^3)](n,k) = T(n+3,k+3)/27^(n-k).
...
(2) Illustrate [T^( 1/3^(n-1) )](n,k) = (3^k)^(n-k) as follows.
Matrix cube root, T^(1/3), begins:
1;
1/3, 1;
1, 3, 1; <== row 2: [T^(1/3^1)](2,k) = (3^k)^(2-k), k=0..2
13, 54, 27, 1;
1083, 5427, 3645, 243, 1;
601329, 3537108, 2919645, 275562, 2187, 1;
2383212465, 16064505711, 15557580981, 1872266643, 21789081, 19683, 1; ...
Matrix 9th root, T^(1/9), begins:
1;
1/9, 1;
2/9, 1, 1;
1, 9, 9, 1; <== row 3: [T^(1/3^2)](3,k) = (3^k)^(3-k), k=0..3
34, 351, 486, 81, 1;
6907, 87723, 146529, 32805, 729, 1;
9623667, 146122947, 286505748, 78830415, 2480058, 6561, 1;
...
Matrix 27th root, T^(1/27), begins:
1;
1/27, 1;
5/81, 1/3, 1;
5/81, 2, 3, 1;
1, 27, 81, 27, 1; <== row 4: [T^(1/3^3)](4,k) = (3^k)^(4-k), k=0..4
193/3, 2754, 9477, 4374, 243, 1;
26497, 1678401, 7105563, 3956283, 295245, 2187, 1;
...
Matrix 81st root, T^(1/81), begins:
1;
1/81, 1;
14/729, 1/9, 1;
-13/2187, 5/9, 1, 1;
-16/243, 5/3, 18, 9, 1;
1, 81, 729, 729, 81, 1; <== row 5: [T^(1/3^4)](5,k) = (3^k)^(5-k), k=0..5
43/3, 15633, 223074, 255879, 39366, 729, 1;
...

Prog

(PARI) {T(n, k)=local(M=Mat(1), L, R); for(i=1, n,
L=sum(j=1, #M, -(M^0-M)^j/j); M=sum(j=0, #L, (L/3^(#L-1))^j/j!); R=matrix(#M+1, #M+1, r, c,
if(r>=c, if(r<=#M, M[r, c], 3^((c-1)*(#M+1-c))))); M=R^(3^(#M-1)) ); M[n+1, k+1]}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A134049, A274481.

Sequence in context: A179593 A117871 A011454 * A115145 A296478 A195403

Adjacent sequences: A274477 A274478 A274479 * A274481 A274482 A274483

Keyword

nonn,tabl

Author

Paul D. Hanna, Jun 24 2016

Status

approved