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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 STATUS approved

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