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Search: id:A111942
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| A111942 |
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Column 0 of the matrix logarithm (A111941) of triangle A111940, which shifts columns left and up under matrix inverse; these terms are the result of multiplying the element in row n by n!. |
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+0
4
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| 0, 1, -1, 1, -2, 4, -12, 36, -144, 576, -2880, 14400, -86400, 518400, -3628800, 25401600, -203212800, 1625702400, -14631321600, 131681894400, -1316818944000, 13168189440000, -144850083840000, 1593350922240000, -19120211066880000
(list; graph; listen)
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OFFSET
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0,5
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COMMENT
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Signed version of A010551, with leading zero.
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FORMULA
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a(n) = (-1)^(n-1) * [(n-1)/2]! * [n/2]! for n>0, with a(0)=0. E.g.f.: A(x) = (1-x/2)/sqrt(1-x^2/4)*acos(1-x^2/2).
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EXAMPLE
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E.g.f.: A(x) = x - 1/2!*x^2 + 1/3!*x^3 - 2/4!*x^4 + 4/5!*x^5
- 12/6!*x^6 + 36/7!*x^7 - 144/8!*x^8 + 576/9!*x^9 +...
where A(x)*A(-x) = -acos(1-x^2/2)^2.
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PROGRAM
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(PARI) {a(n, q=-1)=local(A=Mat(1), B); if(n<0, 0, for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=1, if(j==1, B[i, j]=(A^q)[i-1, 1], B[i, j]=(A^q)[i-1, j-1])); )); A=B); B=sum(i=1, #A, -(A^0-A)^i/i); return(n!*B[n+1, 1]))}
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CROSSREFS
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Cf. A111940 (triangle), A111941 (matrix log), A110505 (variant), A010551 (unsigned).
Adjacent sequences: A111939 A111940 A111941 this_sequence A111943 A111944 A111945
Sequence in context: A062161 A046993 A010551 this_sequence A003701 A114500 A108532
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KEYWORD
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sign
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Aug 23 2005
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