A144442 - OEIS

%I #9 Mar 04 2022 01:28:14

%S 1,1,1,1,17,1,1,118,118,1,1,729,2681,729,1,1,4400,41745,41745,4400,1,

%T 1,26431,555240,1349245,555240,26431,1,1,158622,6816846,33456685,

%U 33456685,6816846,158622,1,1,951773,80034743,715321156,1411926995,715321156,80034743,951773,1

%N 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 G. C. Greubel, <a href="/A144442/b144442.txt">Rows n = 1..50 of the triangle, flattened</a>

%F 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 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 From _G. C. Greubel_, Mar 03 2022: (Start)

%F T(n, n-k) = T(n, k).

%F T(n, 2) = (1/5)*(17*6^(n - 2) - (5*n + 2)).

%F T(n, 3) = (1/50)*(25*n^2 - 5*n - 31 - 34*6^(n - 3)*(30*n - 13) +

%F    2489*11^(n - 3)). (End)

%e Triangle begins as:

%e   1;

%e   1,      1;

%e   1,     17,        1;

%e   1,    118,      118,         1;

%e   1,    729,     2681,       729,          1;

%e   1,   4400,    41745,     41745,       4400,         1;

%e   1,  26431,   555240,   1349245,     555240,     26431,        1;

%e   1, 158622,  6816846,  33456685,   33456685,   6816846,   158622,      1;

%e   1, 951773, 80034743, 715321156, 1411926995, 715321156, 80034743, 951773, 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,5,5], {n,12}, {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 A144442(n,k): return T(n,k,5,5)

%o flatten([[A144442(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, A144441, A144443, A144444, A144445.

%K nonn,tabl

%O 1,5

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 05 2008