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