A144435 - OEIS

%I #9 Mar 03 2022 03:48:15

%S 1,1,1,1,2,1,1,1,1,1,1,0,2,0,1,1,-1,0,0,-1,1,1,-2,3,4,3,-2,1,1,-3,3,

%T -17,-17,3,-3,1,1,-4,8,28,110,28,8,-4,1,1,-5,10,-90,-476,-476,-90,10,

%U -5,1

%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 = 2, read by rows.

%H G. C. Greubel, <a href="/A144435/b144435.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 = 2.

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

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

%F T(n, 2) = 5 - n, n >= 3.

%F T(n, 3) = (1/2)*(n^2 - 11*n + 32) - (-1)^n, n >= 4.

%F T(n, 4) = (1/12)*(-2*n^3 + 36*n^2 - 226*n + 496 - (-1)^n*(2^n - 12*(n-1))), n >= 5. (End)

%e Triangle begins as:

%e   1;

%e   1,  1;

%e   1,  2,  1;

%e   1,  1,  1,   1;

%e   1,  0,  2,   0,    1;

%e   1, -1,  0,   0,   -1,    1;

%e   1, -2,  3,   4,    3,   -2,   1;

%e   1, -3,  3, -17,  -17,    3,  -3,  1;

%e   1, -4,  8,  28,  110,   28,   8, -4,  1;

%e   1, -5, 10, -90, -476, -476, -90, 10, -5, 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,-1,2], {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 A144435(n,k): return T(n,k,-1,2)

%o flatten([[A144435(n,k) for k in (1..n)] for n in (1..15)]) # _G. C. Greubel_, Mar 03 2022

%Y Cf. A144431, A144432, A144436.

%K sign,tabl

%O 1,5

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

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