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A118185 Triangle T, read by rows, defined by: T(n,k) = (4^k)^(n-k) for n>=k>=0. 9
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 (list; table; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..44.

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

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).

PROG

(PARI) T(n, k)=if(n<k || k<0, 0, (4^k)^(n-k) )

CROSSREFS

Cf. A118186 (row sums), A118187 (antidiagonal sums); A118188, A118189; variants: A117401 (q=2), A118180 (q=3), A118190 (q=5).

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

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Last modified March 31 03:44 EDT 2020. Contains 333136 sequences. (Running on oeis4.)