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A144438 Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

%I A144438 #9 Mar 03 2022 17:06:02
%S A144438 1,1,1,1,5,1,1,14,14,1,1,33,89,33,1,1,72,413,413,72,1,1,151,1632,3393,
%T A144438 1632,151,1,1,310,5874,22145,22145,5874,310,1,1,629,19943,125456,
%U A144438 224843,125456,19943,629,1,1,1268,65171,647299,1899096,1899096,647299,65171,1268,1
%N A144438 Triangle T(n,k) by rows: T(n, k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
%H A144438 G. C. Greubel, <a href="/A144438/b144438.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A144438 T(n,k) = (n-k+1)*T(n-1, k-1) + k*T(n-1, k) + T(n-2, k-1), T(n, 1) = T(n, n) = 1.
%F A144438 Sum_{k=1..n} T(n, k) = A001053(n+1).
%F A144438 From _G. C. Greubel_, Mar 03 2022: (Start)
%F A144438 T(n, n-k) = T(n, k).
%F A144438 T(n, 3) = (1/2)*(n^2 +3*n +1 + 73*3^(n-3) - 5*2^(n-2)*(2*n+3)). (End)
%e A144438 Triangle begins as:
%e A144438   1;
%e A144438   1,    1;
%e A144438   1,    5,     1;
%e A144438   1,   14,    14,      1;
%e A144438   1,   33,    89,     33,       1;
%e A144438   1,   72,   413,    413,      72,       1;
%e A144438   1,  151,  1632,   3393,    1632,     151,      1;
%e A144438   1,  310,  5874,  22145,   22145,    5874,    310,     1;
%e A144438   1,  629, 19943, 125456,  224843,  125456,  19943,   629,    1;
%e A144438   1, 1268, 65171, 647299, 1899096, 1899096, 647299, 65171, 1268, 1;
%t A144438 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 A144438 Table[T[n,k,1,1], {n,15}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 03 2022 *)
%o A144438 (Sage)
%o A144438 def T(n,k,m,j):
%o A144438     if (k==1 or k==n): return 1
%o A144438     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 A144438 def A144438(n,k): return T(n,k,1,1)
%o A144438 flatten([[A144438(n,k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Mar 03 2022
%Y A144438 Cf. A001053 (row sums), A094002 (column k=2).
%Y A144438 Cf. A144431, A144432, A144435, A144436.
%K A144438 nonn,tabl
%O A144438 1,5
%A A144438 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 05 2008