A118185 Triangle T(n,k) = 4^(k*(n-k)) for n>=k>=0, read by rows.

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).

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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

Adjacent sequences: A118182 A118183 A118184 * A118186 A118187 A118188

Keyword

nonn,tabl

Author

Paul D. Hanna, Apr 15 2006

Status

approved