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).
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
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Apr 15 2006
STATUS
approved