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A144442 Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

%I A144442 #9 Mar 04 2022 01:28:14
%S A144442 1,1,1,1,17,1,1,118,118,1,1,729,2681,729,1,1,4400,41745,41745,4400,1,
%T A144442 1,26431,555240,1349245,555240,26431,1,1,158622,6816846,33456685,
%U A144442 33456685,6816846,158622,1,1,951773,80034743,715321156,1411926995,715321156,80034743,951773,1
%N A144442 Triangle read by rows: T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
%H A144442 G. C. Greubel, <a href="/A144442/b144442.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A144442 T(n, k) = (5*n-5*k+1)*T(n-1, k-1) +(5*k-4)*T(n-1, k) + 5*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.
%F A144442 Sum_{k=1..n} T(n, k) = s(n), where s(n) = (5*n-8)*s(n-1) + 5*s(n-2), with s(1) = 1, s(2) = 2.
%F A144442 From _G. C. Greubel_, Mar 03 2022: (Start)
%F A144442 T(n, n-k) = T(n, k).
%F A144442 T(n, 2) = (1/5)*(17*6^(n - 2) - (5*n + 2)).
%F A144442 T(n, 3) = (1/50)*(25*n^2 - 5*n - 31 - 34*6^(n - 3)*(30*n - 13) +
%F A144442    2489*11^(n - 3)). (End)
%e A144442 Triangle begins as:
%e A144442   1;
%e A144442   1,      1;
%e A144442   1,     17,        1;
%e A144442   1,    118,      118,         1;
%e A144442   1,    729,     2681,       729,          1;
%e A144442   1,   4400,    41745,     41745,       4400,         1;
%e A144442   1,  26431,   555240,   1349245,     555240,     26431,        1;
%e A144442   1, 158622,  6816846,  33456685,   33456685,   6816846,   158622,      1;
%e A144442   1, 951773, 80034743, 715321156, 1411926995, 715321156, 80034743, 951773, 1;
%t A144442 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 A144442 Table[T[n,k,5,5], {n,12}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 03 2022 *)
%o A144442 (Sage)
%o A144442 def T(n,k,m,j):
%o A144442     if (k==1 or k==n): return 1
%o A144442     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 A144442 def A144442(n,k): return T(n,k,5,5)
%o A144442 flatten([[A144442(n,k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Mar 03 2022
%Y A144442 Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144440, A144441, A144443, A144444, A144445.
%K A144442 nonn,tabl
%O A144442 1,5
%A A144442 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 05 2008