A144436 - OEIS

%I #6 Mar 03 2022 04:43:15

%S 1,1,1,1,8,1,1,23,23,1,1,54,170,54,1,1,117,818,818,117,1,1,244,3255,

%T 7224,3255,244,1,1,499,11697,48443,48443,11697,499,1,1,1010,39560,

%U 276974,513326,276974,39560,1010,1,1,2033,128756,1431604,4422246

%N 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 = 1, and j = 4, read by rows.

%H G. C. Greubel, <a href="/A144436/b144436.txt">Rows n = 1..50 of the triangle, flattened</a>

%F 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 = 1, and j = 4.

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

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

%F T(n, 2) = 2^(n+1) - (n+5).

%F T(n, 3) = (1/2)*( n^2 + 9*n + 16 - 2^(n+2)*(n+3) + 142*3^(n-3) ). (End)

%e Triangle begins as:

%e   1;

%e   1,    1;

%e   1,    8,      1;

%e   1,   23,     23,       1;

%e   1,   54,    170,      54,       1;

%e   1,  117,    818,     818,     117,       1;

%e   1,  244,   3255,    7224,    3255,     244,       1;

%e   1,  499,  11697,   48443,   48443,   11697,     499,      1;

%e   1, 1010,  39560,  276974,  513326,  276974,   39560,   1010,    1;

%e   1, 2033, 128756, 1431604, 4422246, 4422246, 1431604, 128756, 2033, 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,1,4], {n,15}, {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 A144436(n,k): return T(n,k,1,4)

%o flatten([[A144436(n,k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Mar 03 2022

%Y Cf. A144431, A144432, A144435.

%K nonn,tabl

%O 1,5

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 04 2008

%E Edited by _G. C. Greubel_, Mar 03 2022