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A158117 Triangle T(n, k) = 10^(k*(n-k)), read by rows.

%I A158117 #14 Sep 08 2022 08:45:42
%S A158117 1,1,1,1,10,1,1,100,100,1,1,1000,10000,1000,1,1,10000,1000000,1000000,
%T A158117 10000,1,1,100000,100000000,1000000000,100000000,100000,1,1,1000000,
%U A158117 10000000000,1000000000000,1000000000000,10000000000,1000000,1
%N A158117 Triangle T(n, k) = 10^(k*(n-k)), read by rows.
%H A158117 G. C. Greubel, <a href="/A158117/b158117.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A158117 T(n, k, q) = c(n,q)/(c(k,q)*c(n-k,q)) where c(n, k) = binomial(q+2, 2)^binomial(n+1, 2), c(n, 0) = n!, and q = 3.
%F A158117 T(n, k, q) = binomial(q+2, 2)^(k*(n-k)) with q = 3.
%F A158117 T(n, k, m) = (m+2)^(k*(n-k)) with m = 8. - _G. C. Greubel_, Jun 30 2021
%e A158117 Triangle begins as:
%e A158117   1;
%e A158117   1,       1;
%e A158117   1,      10,           1;
%e A158117   1,     100,         100,             1;
%e A158117   1,    1000,       10000,          1000,             1;
%e A158117   1,   10000,     1000000,       1000000,         10000,           1;
%e A158117   1,  100000,   100000000,    1000000000,     100000000,      100000,       1;
%e A158117   1, 1000000, 10000000000, 1000000000000, 1000000000000, 10000000000, 1000000, 1;
%t A158117 (* First program *)
%t A158117 T[n_, k_, q_]= Binomial[q+2,2](k*(n-k));
%t A158117 Table[T[n,k,3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by _G. C. Greubel_, Jun 30 2021 *)
%t A158117 (* Second program *)
%t A158117 With[{m=8}, Table[(m+2)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten] (* _G. C. Greubel_, Jun 30 2021 *)
%o A158117 (Magma) [10^(k*(n-k)): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 30 2021
%o A158117 (Sage) flatten([[10^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 30 2021
%Y A158117 Cf. A007318 (q=0), A118180 (q=1), A158116 (q=2), this sequence (q=3), A176639 (q=4), A176643 (q=5), A176641 (q=6).
%Y A158117 Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), this sequence (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), A176631 (m=20), A176641 (m=26).
%K A158117 nonn,tabl
%O A158117 0,5
%A A158117 _Roger L. Bagula_, Mar 12 2009
%E A158117 Edited by _G. C. Greubel_, Jun 30 2021