A173567 - OEIS

%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