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A045648 Number of chiral n-ominoes in (n-1)-space, one cell labeled. 14
1, 1, 1, 2, 4, 8, 16, 34, 75, 166, 370, 841, 1937, 4488, 10470, 24617, 58237, 138435, 330563, 792745, 1908379, 4609434, 11167781, 27134824, 66102921, 161417867, 395042562, 968791315, 2380383481, 5859176855, 14446043494, 35672895787, 88219204394, 218466647493 (list; graph; refs; listen; history; text; internal format)
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

Cf. A045649, A000081, A004111.

Sequence in context: A006981 A003427 A333647 * A248890 A308245 A209971

Adjacent sequences:  A045645 A045646 A045647 * A045649 A045650 A045651

KEYWORD

easy,nonn

AUTHOR

Robert A. Russell

STATUS

approved

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Last modified June 13 11:07 EDT 2021. Contains 344989 sequences. (Running on oeis4.)