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A118441 Triangle L, read by rows, equal to the matrix log of A118435, with the property that L^2 consists of a single diagonal (two rows down from the main diagonal). 5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

LINKS

Table of n, a(n) for n=0..54.

FORMULA

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).

EXAMPLE

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; ...

PROG

(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]}

for(n=0, 10, for(k=0, n, print1(L(n, k), ", ")); print(""))

(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)))}

for(n=0, 10, for(k=0, n, print1(L(n, k, 1), ", ")); print(""))

CROSSREFS

Cf. A118435 (exp(L)), A118442 (column 0), A118443 (row sums), A027471 (unsigned row sums); A118433 (self-inverse triangle), A001815 (column 1?), A001789 (third of column 2?).

Sequence in context: A219245 A299769 A091435 * A244131 A206428 A111549

Adjacent sequences:  A118438 A118439 A118440 * A118442 A118443 A118444

KEYWORD

sign,tabl

AUTHOR

Paul D. Hanna, Apr 28 2006

STATUS

approved

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Last modified January 17 16:44 EST 2019. Contains 319235 sequences. (Running on oeis4.)