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A110503 Triangle, read by rows, which shifts one column left under matrix inverse. 8
1, 1, 1, 1, -1, 1, 1, -2, 1, 1, 1, -1, 1, -1, 1, 1, -1, 1, -2, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -2, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, 1, -2, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, 1, -1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

The unsigned columns of the matrix logarithm of this triangle are all equal to A110504.

LINKS

Table of n, a(n) for n=0..97.

FORMULA

T(n, k) = +1 when k == 0 (mod 2), T(n, k)=-1 when k == 1 (mod 2), except for T(k+2, k) = -2 when k == 1 (mod 2) and T(n, n) = 1.

G.f. for column k of matrix power A110503^m (ignoring leading zeros): cos(m*arccos(1-x^2/2)) + (-1)^k*sin(m*arccos(1-x^2/2))*(1-x/2)/sqrt(1-x^2/4)*(1+x)/(1-x).

EXAMPLE

Triangle begins:

1;

1, 1;

1, -1, 1;

1, -2, 1, 1;

1, -1, 1, -1, 1;

1, -1, 1, -2, 1, 1;

1, -1, 1, -1, 1, -1, 1;

1, -1, 1, -1, 1, -2, 1, 1;

1, -1, 1, -1, 1, -1, 1, -1, 1;

1, -1, 1, -1, 1, -1, 1, -2, 1, 1; ...

The matrix inverse drops the first column:

1;

-1, 1;

-2, 1, 1;

-1, 1, -1, 1;

-1, 1, -2, 1, 1;

-1, 1, -1, 1, -1, 1; ...

The matrix logarithm equals:

0;

1/1!, 0;

3/2!, -1/1!, 0;

7/3!, -3/2!, 1/1!, 0;

30/4!, -7/3!, 3/2!, -1/1!, 0;

144/5!, -30/4!, 7/3!, -3/2!, 1/1!, 0;

876/6!, -144/5!, 30/4!, -7/3!, 3/2!, -1/1!, 0; ...

unsigned columns of which all equal A110505.

PROG

(PARI) T(n, k)=matrix(n+1, n+1, r, c, if(r>=c, if(r==c || c%2==1, 1, if(r%2==0 && r==c+2, -2, -1))))[n+1, k+1]

CROSSREFS

Cf. A110504 (matrix log), A110505 (column 0 of log).

Cf. A111940 (variant).

Sequence in context: A147300 A346481 A240228 * A030556 A083230 A043284

Adjacent sequences: A110500 A110501 A110502 * A110504 A110505 A110506

KEYWORD

sign,tabl

AUTHOR

Paul D. Hanna, Jul 23 2005

STATUS

approved

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Last modified November 29 13:13 EST 2022. Contains 358429 sequences. (Running on oeis4.)