A104986 Matrix logarithm of triangle A104980.

0, 1, 0, 2, 2, 0, 7, 4, 3, 0, 33, 14, 7, 4, 0, 191, 66, 27, 11, 5, 0, 1297, 382, 137, 48, 16, 6, 0, 10063, 2594, 843, 270, 79, 22, 7, 0, 87669, 20126, 6041, 1820, 495, 122, 29, 8, 0, 847015, 175338, 49219, 14176, 3679, 848, 179, 37, 9, 0, 8989301, 1694030, 448681, 124828, 31361, 6930, 1371, 252, 46, 10, 0

Offset

0,4

Comments

Column 0 equals column 1 of triangular matrix A104980, which satisfies: SHIFT_LEFT(column 0 of A104980^p) = p*(column p+1 of A104980) for p>=0. Column 1 equals twice column 0.

Links

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

Formula

T(n, 0) = A104981(n), T(n+1, 1) = 2*T(n, 0) for n>=0.

Example

Triangle begins:
        0;
        1,       0;
        2,       2,      0;
        7,       4,      3,      0;
       33,      14,      7,      4,     0;
      191,      66,     27,     11,     5,    0;
     1297,     382,    137,     48,    16,    6,    0;
    10063,    2594,    843,    270,    79,   22,    7,   0;
    87669,   20126,   6041,   1820,   495,  122,   29,   8,  0;
   847015,  175338,  49219,  14176,  3679,  848,  179,  37,  9,  0;
  8989301, 1694030, 448681, 124828, 31361, 6930, 1371, 252, 46, 10, 0; ...

Mathematica

nmax = 10;
M = Table[If[n == k, 0, If[n == k+1, -n+1, -Coefficient[(1-1/Sum[i! x^i, {i, 0, n}])/x + O[x]^n, x, n-k-1]]], {n, 1, nmax+1}, {k, 1, nmax+1}];
T[n_, k_] /; 0 <= k <= n := Sum[(-1)^p MatrixPower[M, p][[n+1, k+1]]/p, {p, 1, n+1}]; T[_, _] = 0;
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 09 2018, from PARI *)

Prog

(PARI) T(n, k)=if(n<k || k<0, 0, sum(p=1, n+1, (-1)^p*(matrix(n+1, n+1, m, j, if(m==j, 0, if(m==j+1, -m+1, -polcoeff((1-1/sum(i=0, m, i!*x^i))/x+O(x^m), m-j-1))))^p)[n+1, k+1]/p))

Crossrefs

Cf. A104980, A104981 (column 0), A104987 (row sums).

Sequence in context: A323675 A243492 A086118 * A060007 A021457 A305605

Adjacent sequences: A104983 A104984 A104985 * A104987 A104988 A104989

Keyword

nonn,tabl

Author

Paul D. Hanna, Apr 10 2005

Status

approved