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%I #22 Sep 08 2022 08:45:25
%S 1,1,1,1,5,1,1,25,25,1,1,125,625,125,1,1,625,15625,15625,625,1,1,3125,
%T 390625,1953125,390625,3125,1,1,15625,9765625,244140625,244140625,
%U 9765625,15625,1,1,78125,244140625,30517578125,152587890625,30517578125,244140625,78125,1
%N Triangle T(n,k) = 5^(k*(n-k)) for n >= k >= 0, read by rows.
%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) = A118193(n-k)*T(n,k); also, the matrix log is given by [log(T)](n,k) = A118194(n-k)*T(n,k).
%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/(1-5^n*x).
%H G. C. Greubel, <a href="/A118190/b118190.txt">Rows n = 0..50 of the triangle, flattened</a>
%F G.f.: A(x,y) = Sum_{n>=0} x^n/(1-5^n*x*y).
%F G.f. satisfies: A(x,y) = 1/(1-x*y) + x*A(x,5*y).
%F T(n,k) = (1/n)*( 5^(n-k)*k*T(n-1,k-1) + 5^k*(n-k)*T(n-1,k) ), where T(i,j)=0 if j>i. - _Tom Edgar_, Feb 21 2014
%F T(n, k, m) = (m+2)^(k*(n-k)) with m = 3. - _G. C. Greubel_, Jun 29 2021
%e A(x,y) = 1/(1-x*y) + x/(1-5*x*y) + x^2/(1-25*x*y) + x^3/(1-125*x*y) + ...
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 5, 1;
%e 1, 25, 25, 1;
%e 1, 125, 625, 125, 1;
%e 1, 625, 15625, 15625, 625, 1;
%e 1, 3125, 390625, 1953125, 390625, 3125, 1;
%e 1, 15625, 9765625, 244140625, 244140625, 9765625, 15625, 1; ...
%e The matrix inverse T^-1 starts:
%e 1;
%e -1, 1;
%e 4, -5, 1;
%e -76, 100, -25, 1;
%e 7124, -9500, 2500, -125, 1;
%e -3326876, 4452500, -1187500, 62500, -625, 1; ...
%e where [T^-1](n,k) = A118193(n-k)*(5^k)^(n-k).
%t With[{m=3}, Table[(m+2)^(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, (5^k)^(n-k) )
%o (Magma) [5^(k*(n-k)): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 29 2021
%o (Sage) flatten([[5^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 29 2021
%Y Cf. A118191 (row sums), A118192 (antidiagonal sums), A118193, A118194.
%Y Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), this sequence (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15).
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Apr 15 2006
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