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Search: id:A098777
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| A098777 |
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a(0)=0, a(n+1)=(-1)^(n+1)*sum('binomial(n,k)*a(k)*a(n-k)','k'=0..n), n>=0. |
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+0
1
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| 1, -1, -2, 2, 16, -40, -320, 1040, 12160, -52480, -742400, 3872000, 66457600, -411136000, -8202444800, 58479872000, 1335009280000, -10791497728000, -277035646976000, 2502527565824000
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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A variation of the usual factorials (which satisfy the recursion (n+1)!=sum('binomial(n,k)*k!*(n-k)!','k'=0..n) for n>=0).
This sequence seems to satisfy an analogue of Wilson's Theorem (which states that (p-1)! equals -1 modulo p for p a prime): For p<10000 a prime congruent to 2 modulo 3, we have a(p-1) congruent to 1 mod p and a(n) congruent to 0 mod p for n>p. For p<10000 a prime congruent to 1 mod 3 we have a(p-1)+a(p) congruent to -1 modulo p.
On the analytic side, the sequence is closely related (via its exponential generating series) to the elliptic curve of j-invariant O (corresponding to the regular hexagonal lattice).
The ordinary generating series seems to have a nice continued fraction expansion of Jacobi type.
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FORMULA
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The exponential generating function f(z)=sum('a(n)*z^n/n!', 'n'=0..infinity) satisfies f'(z)=-f(-z)^2 and is an elliptic function with respect to a regular hexagonal lattice (moreover, -f(z)f(-z) is (up to translation) a Weyerstrass function.
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CROSSREFS
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Adjacent sequences: A098774 A098775 A098776 this_sequence A098778 A098779 A098780
Sequence in context: A088139 A113123 A076615 this_sequence A127226 A001119 A062282
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KEYWORD
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sign
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AUTHOR
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Roland Bacher (roland.bacher(AT)ujf-grenoble.fr), Oct 04 2004
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