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%I #22 Sep 08 2022 08:45:25
%S 1,1,1,1,4,1,1,16,16,1,1,64,256,64,1,1,256,4096,4096,256,1,1,1024,
%T 65536,262144,65536,1024,1,1,4096,1048576,16777216,16777216,1048576,
%U 4096,1,1,16384,16777216,1073741824,4294967296,1073741824,16777216,16384,1
%N Triangle T(n,k) = 4^(k*(n-k)) for n>=k>=0, read by rows.
%C 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).
%C 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).
%H G. C. Greubel, <a href="/A118185/b118185.txt">Rows n = 0..50 of the triangle, flattened</a>
%F G.f.: A(x,y) = Sum_{n>=0} x^n/(1-4^n*x*y).
%F G.f. satisfies: A(x,y) = 1/(1-x*y) + x*A(x,4*y).
%F 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
%F T(n, k, m) = (m+2)^(k*(n-k)) with m = 2. - _G. C. Greubel_, Jun 29 2021
%e A(x,y) = 1/(1-xy) + x/(1-4xy) + x^2/(1-16xy) + x^3/(1-64xy) + ...
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 1, 16, 16, 1;
%e 1, 64, 256, 64, 1;
%e 1, 256, 4096, 4096, 256, 1;
%e 1, 1024, 65536, 262144, 65536, 1024, 1;
%e 1, 4096, 1048576, 16777216, 16777216, 1048576, 4096, 1; ...
%e The matrix inverse T^-1 starts:
%e 1;
%e -1, 1;
%e 3, -4, 1;
%e -33, 48, -16, 1;
%e 1407, -2112, 768, -64, 1;
%e -237057, 360192, -135168, 12288, -256, 1; ...
%e where [T^-1](n,k) = A118188(n-k)*4^(k*(n-k)).
%t Table[4^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 29 2021 *)
%o (PARI) T(n, k)=if(n<k || k<0, 0, (4^k)^(n-k) )
%o (Magma) [4^(k*(n-k)): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 29 2021
%o (Sage) flatten([[4^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 29 2021
%Y Cf. A118186 (row sums), A118187 (antidiagonal sums), A118188, A118189.
%Y 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).
%Y T(2n,n) gives A060757.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Apr 15 2006
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