A144443 - OEIS

%I #8 Mar 04 2022 17:17:10

%S 1,1,1,1,20,1,1,159,159,1,1,1138,4254,1138,1,1,7997,77878,77878,7997,

%T 1,1,56016,1219167,2984888,1219167,56016,1,1,392155,17633649,87659315,

%U 87659315,17633649,392155,1,1,2745134,244083268,2219485106,4400875078,2219485106,244083268,2745134,1

%N Triangle read by rows: T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

%H G. C. Greubel, <a href="/A144443/b144443.txt">Rows n = 1..50 of the triangle, flattened</a>

%F T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*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*(3*n-5)*s(n-1) + 6*s(n-2), s(1) = 1, s(2) = 2.

%F From _G. C. Greubel_, Mar 04 2022: (Start)

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

%F T(n, 2) = (1/3)*(10*7^(n-2) - (3*n+1)).

%F T(n, 3) = (1/18)*(9*n^2 -3*n -11 - 20*(21*n-11)*7^(n-3) + 997*13^(n-3)). (End)

%e Triangle begins as:

%e   1;

%e   1,      1;

%e   1,     20,        1;

%e   1,    159,      159,        1;

%e   1,   1138,     4254,     1138,        1;

%e   1,   7997,    77878,    77878,     7997,        1;

%e   1,  56016,  1219167,  2984888,  1219167,    56016,      1;

%e   1, 392155, 17633649, 87659315, 87659315, 17633649, 392155, 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,6,6], {n,15}, {k,n}]//Flatten (* modified by _G. C. Greubel_, Mar 04 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 A144443(n,k): return T(n,k,6,6)

%o flatten([[A144443(n,k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Mar 04 2022

%Y Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144440, A144441, A144442, A144444, A144445.

%K nonn,tabl

%O 1,5

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