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A045648 Number of chiral n-ominoes in (n-1)-space, one cell labeled.

%I A045648 #28 Jun 06 2019 02:41:23
%S A045648 1,1,1,2,4,8,16,34,75,166,370,841,1937,4488,10470,24617,58237,138435,
%T A045648 330563,792745,1908379,4609434,11167781,27134824,66102921,161417867,
%U A045648 395042562,968791315,2380383481,5859176855,14446043494,35672895787,88219204394,218466647493
%N A045648 Number of chiral n-ominoes in (n-1)-space, one cell labeled.
%C A045648 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.
%C A045648 Euler transform of a(n) - if(n%4!=2, 0, a(n/2)) is sequence itself with offset 0.
%D A045648 D. E. Knuth, Fundamental Algorithms, 3d Ed. 1997, pp. 386-388.
%H A045648 Alois P. Heinz, <a href="/A045648/b045648.txt">Table of n, a(n) for n = 1..1000</a>
%H A045648 W. F. Lunnon, <a href="https://doi.org/10.1093/comjnl/18.4.366">Counting Multidimensional Polyominoes</a>, Computer Journal, Vol. 18 (1975), pp. 366-367.
%F A045648 G.f.: A(x) = x exp(A(x) + A(-x^2)/2 + A(x^3)/3 + A(-x^4)/4 + ...).
%F A045648 Also A(x) = Sum_{n >= 1} a(n)*x^n = x / Product_{n >= 1} (1-(-x)^n)^((-1)^n*a(n)).
%F A045648 G.f.: x*Product_{n>0} (1-x^(4n-2))^a(2n-1)/(1-x^n)^a(n).
%F A045648 a(n) ~ c * d^n / n^(3/2), where d = 2.58968405406171542574769690513208346256... and c = 0.386431095907583923297618874742... . - _Vaclav Kotesovec_, Feb 29 2016
%p A045648 with(numtheory):
%p A045648 b:= proc(n) option remember; `if`(n=0, 1, add(add(d*(a(d)-
%p A045648       `if`(irem(d, 4)=2, a(d/2), 0)), d=divisors(j))*b(n-j), j=1..n)/n)
%p A045648     end:
%p A045648 a:= n-> b(n-1):
%p A045648 seq(a(n), n=1..40);  # _Alois P. Heinz_, Feb 24 2015
%t A045648 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} ]
%o A045648 (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 */
%Y A045648 Cf. A045649, A000081, A004111.
%K A045648 easy,nonn
%O A045648 1,4
%A A045648 _Robert A. Russell_