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Search: id:A110503
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| A110503 |
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Triangle, read by rows, which shifts one column left under matrix inverse. |
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+0
8
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| 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; listen)
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OFFSET
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0,8
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COMMENT
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The unsigned columns of the matrix logarithm of this triangle are all equal to A110504.
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FORMULA
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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*acos(1-x^2/2)) + (-1)^k*sin(m*acos(1-x^2/2))*(1-x/2)/sqrt(1-x^2/4)*(1+x)/(1-x).
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EXAMPLE
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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.
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PROGRAM
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(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]}
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CROSSREFS
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Cf. A110504 (matrix log), A110505 (column 0 of log).
Cf. A111940 (variant).
Sequence in context: A086251 A092931 A147300 this_sequence A030556 A030575 A043284
Adjacent sequences: A110500 A110501 A110502 this_sequence A110504 A110505 A110506
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KEYWORD
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sign,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Jul 23 2005
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