%I #9 Mar 04 2022 01:28:04
%S 1,1,1,1,14,1,1,83,83,1,1,432,1550,432,1,1,2181,19898,19898,2181,1,1,
%T 10930,217887,523548,217887,10930,1,1,54679,2199237,10589795,10589795,
%U 2199237,54679,1,1,273428,21203828,184722860,362147222,184722860,21203828,273428,1
%N Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
%H G. C. Greubel, <a href="/A144441/b144441.txt">Rows n = 1..50 of the triangle, flattened</a>
%F T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
%F Sum_{k=1..n} T(n, k) = s(n), where s(n) = 2*(2*n-3)*s(n-1) + 4*s(n-2) with s(1) = 1, s(2) = 2.
%F From _G. C. Greubel_, Mar 03 2022: (Start)
%F T(n, n-k) = T(n, k).
%F T(n, 2) = (1/2)*(7*5^(n-2) - (2*n+1)).
%F T(n, 3) = (1/8)*(4*n^2 - 5 - 14*(10*n-3)*5^(n-3) + 355*9^(n-3)). (End)
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 14, 1;
%e 1, 83, 83, 1;
%e 1, 432, 1550, 432, 1;
%e 1, 2181, 19898, 19898, 2181, 1;
%e 1, 10930, 217887, 523548, 217887, 10930, 1;
%e 1, 54679, 2199237, 10589795, 10589795, 2199237, 54679, 1;
%e 1, 273428, 21203828, 184722860, 362147222, 184722860, 21203828, 273428, 1;
%t T[n_, k_, m_, j_]:= T[n,k,m,j]= If[k==1 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m,j] + (m*(k-1)+1)*T[n-1,k,m,j] + j*T[n-2,k-1,m,j]];
%t Table[T[n,k,4,4], {n,15}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 03 2022 *)
%o (Sage)
%o def T(n,k,m,j):
%o if (k==1 or k==n): return 1
%o else: return (m*(n-k)+1)*T(n-1,k-1,m,j) + (m*(k-1)+1)*T(n-1,k,m,j) + j*T(n-2,k-1,m,j)
%o def A144441(n,k): return T(n,k,4,4)
%o flatten([[A144441(n,k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Mar 03 2022
%Y Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144440, A144442, A144443, A144444, A144445.
%K nonn,tabl
%O 1,5
%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 05 2008