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Search: id:A113278
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| A113278 |
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Triangle T, read by rows, such that the matrix square, T^2, forms a simple 2-diagonal triangle where [T^2](n,n) = 1 and [T^2](n+1,n) = 2*(n+1) for n>=0. |
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+0
2
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| 1, 1, 1, -1, 2, 1, 3, -3, 3, 1, -15, 12, -6, 4, 1, 105, -75, 30, -10, 5, 1, -945, 630, -225, 60, -15, 6, 1, 10395, -6615, 2205, -525, 105, -21, 7, 1, -135135, 83160, -26460, 5880, -1050, 168, -28, 8, 1, 2027025, -1216215, 374220, -79380, 13230, -1890, 252, -36, 9, 1
(list; table; graph; listen)
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OFFSET
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0,5
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EXAMPLE
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Triangle begins:
1;
1,1;
-1,2,1;
3,-3,3,1;
-15,12,-6,4,1;
105,-75,30,-10,5,1;
-945,630,-225,60,-15,6,1;
10395,-6615,2205,-525,105,-21,7,1; ...
where T(n,k) = (-1)^(n-1-k)*A001147(n-1)*C(n,k)
and A001147 forms the odd double factorials.
The matrix square equals:
1;
2,1;
0,4,1;
0,0,6,1;
0,0,0,8,1;
0,0,0,0,10,1;
0,0,0,0,0,12,1; ...
The matrix log, L, begins:
0;
1,0;
-2,2,0;
8,-6,3,0;
-48,32,-12,4,0;
384,-240,80,-20,5,0;
-3840,2304,-720,160,-30,6,0; ...
where L(n,k) = (-1)^(n-1-k)*A000165(n-1)*C(n,k)
and A000165 forms the even double factorials.
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PROGRAM
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(PARI) {T(n, k)=local(M=matrix(n+1, n+1, r, c, if(r==c, 1, if(r==c+1, 2*c)))); (sum(i=0, n+1, (sum(j=1, n+1, -(M^0-M)^j/j)/2)^i/i!))[n+1, k+1]}
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CROSSREFS
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Sequence in context: A039775 A136018 A138022 this_sequence A132382 A048865 A058754
Adjacent sequences: A113275 A113276 A113277 this_sequence A113279 A113280 A113281
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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), Oct 22 2005
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