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