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A104986 Matrix logarithm of triangle A104980.

%I A104986 #15 Jun 08 2021 14:58:18
%S A104986 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,
%T A104986 6,0,10063,2594,843,270,79,22,7,0,87669,20126,6041,1820,495,122,29,8,
%U A104986 0,847015,175338,49219,14176,3679,848,179,37,9,0,8989301,1694030,448681,124828,31361,6930,1371,252,46,10,0
%N A104986 Matrix logarithm of triangle A104980.
%C A104986 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.
%H A104986 G. C. Greubel, <a href="/A104986/b104986.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A104986 T(n, 0) = A104981(n), T(n+1, 1) = 2*T(n, 0) for n>=0.
%e A104986 Triangle begins:
%e A104986         0;
%e A104986         1,       0;
%e A104986         2,       2,      0;
%e A104986         7,       4,      3,      0;
%e A104986        33,      14,      7,      4,     0;
%e A104986       191,      66,     27,     11,     5,    0;
%e A104986      1297,     382,    137,     48,    16,    6,    0;
%e A104986     10063,    2594,    843,    270,    79,   22,    7,   0;
%e A104986     87669,   20126,   6041,   1820,   495,  122,   29,   8,  0;
%e A104986    847015,  175338,  49219,  14176,  3679,  848,  179,  37,  9,  0;
%e A104986   8989301, 1694030, 448681, 124828, 31361, 6930, 1371, 252, 46, 10, 0; ...
%t A104986 nmax = 10;
%t A104986 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 A104986 T[n_, k_] /; 0 <= k <= n := Sum[(-1)^p MatrixPower[M, p][[n+1, k+1]]/p, {p, 1, n+1}]; T[_, _] = 0;
%t A104986 Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Aug 09 2018, from PARI *)
%o A104986 (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))
%Y A104986 Cf. A104980, A104981 (column 0), A104987 (row sums).
%K A104986 nonn,tabl
%O A104986 0,4
%A A104986 _Paul D. Hanna_, Apr 10 2005