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%I #9 Apr 26 2021 01:55:09
%S 2,5,5,9,30,9,14,123,123,14,20,425,1092,425,20,27,1413,7650,7650,1413,
%T 27,35,4872,54051,87380,54051,4872,35,44,17783,426573,943190,943190,
%U 426573,17783,44,54,67875,3655854,12192579,12207030,12192579,3655854,67875,54
%N Triangle T(n, k) = (f(k, n-k+1) + f(n-k+1, k))/2 where f(n, k) = (1/2)*Sum_{j=1..2*n} k^j, read by rows.
%H G. C. Greubel, <a href="/A173567/b173567.txt">Rows n = 1..50 of the triangle, flattened</a>
%F T(n, k) = (f(k, n-k+1) + f(n-k+1, k))/2 where f(n, k) = (1/2)*Sum_{j=1..2*n} k^j.
%F T(n, k) = (f(k, n-k+1) + f(n-k+1, k))/2 where f(n, k) = k*(1 - k^(2*n))/(1-k) with f(n, 1) = 2*n. - _G. C. Greubel_, Apr 25 2021
%e Triangle begins as:
%e 2;
%e 5, 5;
%e 9, 30, 9;
%e 14, 123, 123, 14;
%e 20, 425, 1092, 425, 20;
%e 27, 1413, 7650, 7650, 1413, 27;
%e 35, 4872, 54051, 87380, 54051, 4872, 35;
%e 44, 17783, 426573, 943190, 943190, 426573, 17783, 44;
%e 54, 67875, 3655854, 12192579, 12207030, 12192579, 3655854, 67875, 54;
%t f[n_, k_]:= If[k==1, 2*n, k*(1-k^(2*n))/(1-k)];
%t T[n_, k_]:= (f[k, n-k+1] + f[n-k+1, k])/2;
%t Table[T[n, k], {n,10}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Apr 25 2021 *)
%o (Sage)
%o def f(n,k): return 2*n if k==1 else k*(1-k^(2*n))/(1-k)
%o def T(n,k): return (f(k, n-k+1) + f(n-k+1, k))/2
%o flatten([[T(n,k) for k in (1..n)] for n in (1..10)]) # _G. C. Greubel_, Apr 25 2021
%K nonn,tabl
%O 1,1
%A _Roger L. Bagula_, Feb 22 2010
%E Edited by _G. C. Greubel_, Apr 25 2021
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