A117398 Matrix log of triangle A117396.

0, 1, 0, 0, 2, 0, -1, 2, 3, 0, -3, 4, 5, 4, 0, -9, 14, 15, 9, 5, 0, -33, 68, 65, 34, 14, 6, 0, -153, 404, 359, 174, 63, 20, 7, 0, -873, 2804, 2375, 1098, 371, 104, 27, 8, 0, -5913, 22244, 18215, 8154, 2639, 692, 159, 35, 9, 0, -46233, 198644, 158615, 69354, 21791, 5480, 1179, 230, 44, 10, 0

Offset: 0

Author

Paul D. Hanna, Mar 11 2006

Keywords

• sign
• tabl

Comments

Column 0 contains negative of sequence A007489.

Examples

			Triangle begins:
      0;
      1,     0;
      0,     2,     0;
     -1,     2,     3,    0;
     -3,     4,     5,    4,    0;
     -9,    14,    15,    9,    5,   0;
    -33,    68,    65,   34,   14,   6,   0;
   -153,   404,   359,  174,   63,  20,   7,  0;
   -873,  2804,  2375, 1098,  371, 104,  27,  8,  0;
  -5913, 22244, 18215, 8154, 2639, 692, 159, 35,  9,  0;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A117396, A007489 (column 0), A117399 (column 1).

Cf. A000027, A000096, A003422.

Programs

Mathematica

m=12;
M= Table[If[k>n-1, 0, If[k==n-1, n, -1]], {n,0,m+1}, {k,0,m+1}];
T:= T= Sum[MatrixPower[M, j]/j, {j,m+1}];
Table[T[[n+1, k+1]], {n,0,m}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 06 2022 *)

PARI

{T(n,k)=local(M=matrix(n+4,n+4,r,c,if(r>=c,if(r==c+1,-c,1))), L=sum(m=1,n+4,(M^0-M)^m/m));L[n+1,k+1]}

Formula

From G. C. Greubel, Sep 06 2022: (Start)

T(n, n) = 0.

T(n, n-1) = A000027(n).

T(n, n-2) = A000096(n-2).

T(n, 0) = n*[n<2] - A007489(n-2)*[n>1].

T(n, 1) = 0 + 2*A117399(n-1)*[n>1].

Sum_{k=0..n} T(n, k) = A003422(n). (End)