0,3

A variation on 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).

This sequence has a generating function expressible in terms of the Dixon elliptic function sm(x,0) whose coefficients are A104133. The ordinary generating function has a continued fraction expansion of Jacobi type: the numerators are -j^2*(2-(-1)^j) and the denominators are (-1)^(j-1)(j+1/2+(-1)^j/2). - Philippe.Flajolet(AT)inria.fr and Roland.Bacher(AT)ujf-grenoble.fr, Jan 18 2009

T. D. Noe, Table of n, a(n) for n=0..100

R. Bacher and P. Flajolet, Pseudo-factorials, Elliptic Functions and Continued Fractions, arXiv 0901.1379.

P. Flajolet, Research Papers

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.

a(n) = -n!/R^(n+1)*sum(b^(8*p+4*q)/((p-1/2)*b+(q-1/2)/b)^(n+1), p = -infinity..infinity, q = -infinity..infinity), where b = exp(I*Pi/6) and R = 2^(-4/3)/Pi*GAMMA(1/3)^3. - Philippe.Flajolet(AT)inria.fr and Roland.Bacher(AT)ujf-grenoble.fr, Jan 18 2009

Cf. A144689, A144750.

Sequence in context: A152556 A113123 A076615 this_sequence A127226 A001119 A062282

Adjacent sequences: A098774 A098775 A098776 this_sequence A098778 A098779 A098780

sign

Roland Bacher (roland.bacher(AT)ujf-grenoble.fr), Oct 04 2004