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A117398 Matrix log of triangle A117396.

%I A117398 #8 Sep 07 2022 04:08:09
%S A117398 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,
%T A117398 -153,404,359,174,63,20,7,0,-873,2804,2375,1098,371,104,27,8,0,-5913,
%U A117398 22244,18215,8154,2639,692,159,35,9,0,-46233,198644,158615,69354,21791,5480,1179,230,44,10,0
%N A117398 Matrix log of triangle A117396.
%C A117398 Column 0 contains negative of sequence A007489.
%H A117398 G. C. Greubel, <a href="/A117398/b117398.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A117398 From _G. C. Greubel_, Sep 06 2022: (Start)
%F A117398 T(n, n) = 0.
%F A117398 T(n, n-1) = A000027(n).
%F A117398 T(n, n-2) = A000096(n-2).
%F A117398 T(n, 0) = n*[n<2] - A007489(n-2)*[n>1].
%F A117398 T(n, 1) = 0 + 2*A117399(n-1)*[n>1].
%F A117398 Sum_{k=0..n} T(n, k) = A003422(n). (End)
%e A117398 Triangle begins:
%e A117398       0;
%e A117398       1,     0;
%e A117398       0,     2,     0;
%e A117398      -1,     2,     3,    0;
%e A117398      -3,     4,     5,    4,    0;
%e A117398      -9,    14,    15,    9,    5,   0;
%e A117398     -33,    68,    65,   34,   14,   6,   0;
%e A117398    -153,   404,   359,  174,   63,  20,   7,  0;
%e A117398    -873,  2804,  2375, 1098,  371, 104,  27,  8,  0;
%e A117398   -5913, 22244, 18215, 8154, 2639, 692, 159, 35,  9,  0;
%t A117398 m=12;
%t A117398 M= Table[If[k>n-1, 0, If[k==n-1, n, -1]], {n,0,m+1}, {k,0,m+1}];
%t A117398 T:= T= Sum[MatrixPower[M, j]/j, {j,m+1}];
%t A117398 Table[T[[n+1, k+1]], {n,0,m}, {k,0,n}]//Flatten (* _G. C. Greubel_, Sep 06 2022 *)
%o A117398 (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]}
%Y A117398 Cf. A117396, A007489 (column 0), A117399 (column 1).
%Y A117398 Cf. A000027, A000096, A003422.
%K A117398 sign,tabl
%O A117398 0,5
%A A117398 _Paul D. Hanna_, Mar 11 2006