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A144439 Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 2, and j = 2, read by rows.

%I A144439 #9 Mar 10 2022 03:20:08
%S A144439 1,1,1,1,8,1,1,31,31,1,1,102,326,102,1,1,317,2406,2406,317,1,1,964,
%T A144439 15087,34336,15087,964,1,1,2907,86673,380947,380947,86673,2907,1,1,
%U A144439 8738,473084,3650206,6925718,3650206,473084,8738,1,1,26233,2502304,31874880,103245622,103245622,31874880,2502304,26233,1
%N A144439 Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 2, and j = 2, read by rows.
%H A144439 G. C. Greubel, <a href="/A144439/b144439.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A144439 T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 2, and j = 2.
%F A144439 Sum_{k=0..n} T(n, k) = s(n), where s(n) = 2*(n-1)*s(n-1) + 2*s(n-2), s(1) = 1, s(2) = 2.
%F A144439 From _G. C. Greubel_, Mar 10 2022: (Start)
%F A144439 T(n, n-k) = T(n, k).
%F A144439 T(n, 2) = 4*3^(n-2) - (n+1).
%F A144439 T(n, 3) = (1/2)*(71*5^(n-3) - 8*(3*n+1)*3^(n-3) + n^2 + n - 1). (End)
%e A144439 Triangle begins as:
%e A144439   1;
%e A144439   1,     1;
%e A144439   1,     8,       1;
%e A144439   1,    31,      31,        1;
%e A144439   1,   102,     326,      102,         1;
%e A144439   1,   317,    2406,     2406,       317,         1;
%e A144439   1,   964,   15087,    34336,     15087,       964,        1;
%e A144439   1,  2907,   86673,   380947,    380947,     86673,     2907,       1;
%e A144439   1,  8738,  473084,  3650206,   6925718,   3650206,   473084,    8738,     1;
%e A144439   1, 26233, 2502304, 31874880, 103245622, 103245622, 31874880, 2502304, 26233, 1;
%t A144439 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 A144439 Table[T[n,k,2,2], {n,15}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 10 2022 *)
%o A144439 (Sage)
%o A144439 def T(n,k,m,j):
%o A144439     if (k==1 or k==n): return 1
%o A144439     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 A144439 def A144439(n,k): return T(n,k,2,2)
%o A144439 flatten([[A144439(n,k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Mar 10 2022
%Y A144439 Cf. A144431, A144432, A144435, A144436, A144438, A144440, A144441, A144442, A144443, A144444, A144445, A144446, A144447.
%K A144439 nonn,tabl
%O A144439 1,5
%A A144439 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 05 2008