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A118185
Triangle T(n,k) = 4^(k*(n-k)) for n>=k>=0, read by rows.
20
1, 1, 1, 1, 4, 1, 1, 16, 16, 1, 1, 64, 256, 64, 1, 1, 256, 4096, 4096, 256, 1, 1, 1024, 65536, 262144, 65536, 1024, 1, 1, 4096, 1048576, 16777216, 16777216, 1048576, 4096, 1, 1, 16384, 16777216, 1073741824, 4294967296, 1073741824, 16777216, 16384, 1
OFFSET
0,5
COMMENTS
For any column vector C, the matrix product of T*C transforms the g.f. of C: Sum_{n>=0} c(n)*x^n into the g.f.: Sum_{n>=0} c(n)*x^n/(1-4^n*x).
Matrix power T^m satisfies: [T^m](n,k) = [T^m](n-k,0)*T(n,k) for all m and so the triangle has an invariant character. For example, the matrix inverse is defined by [T^-1](n,k) = A118188(n-k)*T(n,k); also, the matrix log is given by [log(T)](n,k) = A118189(n-k)*T(n,k).
FORMULA
G.f.: A(x,y) = Sum_{n>=0} x^n/(1-4^n*x*y).
G.f. satisfies: A(x,y) = 1/(1-x*y) + x*A(x,4*y).
T(n,k) = (1/n)*( 4^(n-k)*k*T(n-1,k-1) + 4^k*(n-k)*T(n-1,k) ), where T(i,j)=0 if j>i. - Tom Edgar, Feb 20 2014
T(n, k, m) = (m+2)^(k*(n-k)) with m = 2. - G. C. Greubel, Jun 29 2021
EXAMPLE
A(x,y) = 1/(1-xy) + x/(1-4xy) + x^2/(1-16xy) + x^3/(1-64xy) + ...
Triangle begins:
1;
1, 1;
1, 4, 1;
1, 16, 16, 1;
1, 64, 256, 64, 1;
1, 256, 4096, 4096, 256, 1;
1, 1024, 65536, 262144, 65536, 1024, 1;
1, 4096, 1048576, 16777216, 16777216, 1048576, 4096, 1; ...
The matrix inverse T^-1 starts:
1;
-1, 1;
3, -4, 1;
-33, 48, -16, 1;
1407, -2112, 768, -64, 1;
-237057, 360192, -135168, 12288, -256, 1; ...
where [T^-1](n,k) = A118188(n-k)*4^(k*(n-k)).
MATHEMATICA
Table[4^(k*(n-k)), {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 29 2021 *)
PROG
(PARI) T(n, k)=if(n<k || k<0, 0, (4^k)^(n-k) )
(Magma) [4^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 29 2021
(Sage) flatten([[4^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 29 2021
CROSSREFS
Cf. A118186 (row sums), A118187 (antidiagonal sums), A118188, A118189.
Cf. A117401 (m=0), A118180 (m=1), this sequence (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15).
T(2n,n) gives A060757.
Sequence in context: A152571 A008304 A203846 * A176483 A174639 A173814
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Apr 15 2006
STATUS
approved