0,4

L = LOG(A118435) = LOG(H*[C^-1]*H], where C=Pascal's triangle and H=A118433 where H^2 = I (identity matrix).

For even exponents of L, L^(2m) is a single diagonal: if n == k+2m, then [L^(2m)](n,k) = n!/k!*2^(n-k-2m)/(n-k-2m)!; else if n != k+2m: [L^(2m)](n,k) = 0. For odd exponents of L: if n >= k+2m+1, then [L^(2m+1)](n,k) = n!/k!*2^(n-k-2m-1)/(n-k-2m-1)!*(-1)^(m+[(n+1)/2]-[k/2]+n-k); else if n < k+2m+1: [L^(2m)](n,k) = 0. Unsigned row sums equals A027471(n+1) = n*3^(n-1).

The matrix log, L = LOG(H*[C^-1]*H], begins:

0;

1, 0;

-4, 2, 0;

-12, 12, 3, 0;

32,-48,-24, 4, 0;

80,-160,-120, 40, 5, 0;

-192, 480, 480,-240,-60, 6, 0;

-448, 1344, 1680,-1120,-420, 84, 7, 0;

1024,-3584,-5376, 4480, 2240,-672,-112, 8, 0;

2304,-9216,-16128, 16128, 10080,-4032,-1008, 144, 9, 0; ...

The matrix square, L^2, is a single diagonal:

0;

0,0;

2,0,0;

0,6,0,0;

0,0,12,0,0;

0,0,0,20,0,0;

0,0,0,0,30,0,0; ...

(PARI) /* From definition of L as matrix log of H*C^-1*H: */ {L(n, k)=local(H=matrix(n+1, n+1, r, c, if(r>=c, binomial(r-1, c-1)*(-1)^(r\2-(c-1)\2+r-c))), C=matrix(n+1, n+1, r, c, if(r>=c, binomial(r-1, c-1))), N=(H*C^-1*H)); Log=sum(p=1, n+1, -(N^0-N)^p/p); Log[n+1, k+1]} (PARI) /* The matrix power L^m is given by: */ {L(n, k, m)=if(m%2==0, if(n==k+m, n!/k!*2^(n-k-m)/(n-k-m)!), if(n>=k+m, n!/k!*2^(n-k-m)/(n-k-m)!*(-1)^(m\2+(n+1)\2-k\2+n-k)))}

Cf. A118435 (exp(L)), A118442 (column 0), A118443 (row sums), A027471 (unsigned row sums); A118433 (self-inverse triangle).

Sequence in context: A135730 A058546 A091435 this_sequence A111549 A022696 A019155

Adjacent sequences: A118438 A118439 A118440 this_sequence A118442 A118443 A118444

sign,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Apr 28 2006