OFFSET
1,4
COMMENTS
Needed for generating chiral n-ominoes in (n-1)-space with no cells labeled, Lunnon's DR(n, n-1) - DE(n, n-1). Knuth describes a method for a similar enumeration, that of free trees with n nodes.
Euler transform of a(n) - if(n%4!=2, 0, a(n/2)) is sequence itself with offset 0.
REFERENCES
D. E. Knuth, Fundamental Algorithms, 3d Ed. 1997, pp. 386-388.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
W. F. Lunnon, Counting Multidimensional Polyominoes, Computer Journal, Vol. 18 (1975), pp. 366-367.
FORMULA
G.f.: A(x) = x exp(A(x) + A(-x^2)/2 + A(x^3)/3 + A(-x^4)/4 + ...).
Also A(x) = Sum_{n >= 1} a(n)*x^n = x / Product_{n >= 1} (1-(-x)^n)^((-1)^n*a(n)).
G.f.: x*Product_{n>0} (1-x^(4n-2))^a(2n-1)/(1-x^n)^a(n).
a(n) ~ c * d^n / n^(3/2), where d = 2.58968405406171542574769690513208346256... and c = 0.386431095907583923297618874742... . - Vaclav Kotesovec, Feb 29 2016
MAPLE
with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*(a(d)-
`if`(irem(d, 4)=2, a(d/2), 0)), d=divisors(j))*b(n-j), j=1..n)/n)
end:
a:= n-> b(n-1):
seq(a(n), n=1..40); # Alois P. Heinz, Feb 24 2015
MATHEMATICA
s[ n_, k_ ] := s[ n, k ]=c[ n+1-k ]+If[ n<2k, 0, s[ n-k, k ](-1)^k ]; c[ 1 ]=1; c[ n_ ] := c[ n ]=Sum[ c[ i ]s[ n-1, i ]i, {i, 1, n-1} ]/(n-1); Table[ c[ i ], {i, 1, 30} ]
PROG
(PARI) {a(n)=local(A=x); if(n<1, 0, for(k=1, n-1, A/=(1-(-x)^k+x*O(x^n))^((-1)^k*polcoeff(A, k))); polcoeff(A, n))} /* Michael Somos, Dec 16 2002 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
STATUS
approved