0,5

Row sums are zero after the initial row. Absolute row sums equal A106339.

Also, T(n, k) = k!*A106340(n, k), where A106340 is the matrix inverse of the triangle formed from (n-k)!*A008278(n, k), n>=k>=0 and A008278 is the triangle of Stirling numbers of 2nd kind.

Triangle begins:

1;

1,-1;

1,-3,2;

1,-9,14,-6;

1,-45,110,-90,24;

1,-585,1670,-1710,744,-120;

1,-21105,61670,-66150,32424,-7560,720;

1,-1858185,5439350,-5864670,2925384,-728280,91440,-5040; ...

The matrix inverse T^-1 begins:

1;

1,1;

1,3/2,1/2;

1,2,7/6,1/6;

1,5/2,25/12,5/8,1/24;

1,3,13/4,3/2,31/120,1/120;

1,7/2,14/3,35/12,301/360,7/80,1/720; ...

where [T^-1](n,k) = A075263(n,k)/n!.

Each row n of the matrix inverse equals the initial

(n+1) fractional coefficients of (x/log(1+x))^n,

which are listed below for n=1,2,3,...,9:

1; 1/2,-1/12,1/24,-19/720,3/160,-863/60480,275/24192,...

1,1; 1/12,0,-1/240,1/240,-221/60480,19/6048,...

1,3/2,1/2; 0,1/240,-1/480,1/945,-11/20160,47/172800,...

1,2,7/6,1/6; -1/720,0,1/3024,-1/3024,199/725760,...

1,5/2,25/12,5/8,1/24; 0,-1/6048,1/12096,-19/725760,...

1,3,13/4,3/2,31/120,1/120; 1/30240,0,-1/57600,1/57600,...

1,7/2,14/3,35/12,301/360,7/80,1/720; 0,1/172800,...

1,4,19/3,5,81/40,23/60,127/5040,1/5040; -1/1209600,0,...

1,9/2,33/4,63/8,331/80,37/32,605/4032,17/2688,1/40320; 0,...

(PARI) {T(n, k)=(M=matrix(n+1, n+1, m, j, if(m>=j, polcoeff((-x/log(1-x+x^2*O(x^n)))^m, j-1)))^-1)[n+1, k+1]} (PARI) {T(n, k)=(-1)^n*k!*(matrix(n+1, n+1, r, c, if(r>=c, (r-c)!* sum(m=0, r-c+1, (-1)^(r-c+1-m)*m^r/m!/(r-c+1-m)!)))^-1)[n+1, k+1]}

Cf. A075263, A106340, A106339, A008278, A002206.

Sequence in context: A119421 A121581 A162976 this_sequence A129964 A100100 A102472

Adjacent sequences: A106335 A106336 A106337 this_sequence A106339 A106340 A106341

sign,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), May 01 2005