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

0,5

Comments

Column 0 contains negative of sequence A007489.

Links

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

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)

Example

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;

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 *)

Prog

(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]}

Crossrefs

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

Cf. A000027, A000096, A003422.

Sequence in context: A308625 A221469 A350369 * A295989 A240852 A363612

Adjacent sequences: A117395 A117396 A117397 * A117399 A117400 A117401

Keyword

sign,tabl

Author

Paul D. Hanna, Mar 11 2006

Status

approved