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A226443 Number of distinct shadow transforms for sequences of length n.

%I A226443 #19 Sep 17 2019 02:53:19
%S A226443 1,1,1,3,12,48,288,1356,10848,70896,588480
%N A226443 Number of distinct shadow transforms for sequences of length n.
%H A226443 Lorenz Halbeisen and Norbert Hungerbuehler, <a href="http://math.berkeley.edu/~halbeis/publications/pdf/seq.pdf">Number theoretic aspects of a combinatorial function</a>, Notes on Number Theory and Discrete Mathematics 5 (1999), 138-150.
%H A226443 Lorenz Halbeisen, <a href="http://www.iam.fmph.uniba.sk/amuc/_vol-74/_no_2/_halbeisen/halbeisen.html">A number-theoretic conjecture and its implication for set theory</a>, Acta Math. Univ. Comenianae 74(2) (2005), 243-254.
%H A226443 OEIS Wiki, <a href="https://oeis.org/wiki/Shadow_transform">Shadow transform</a>.
%F A226443 a(p+1) = (p+1)a(p) where p is prime.
%F A226443 a(n-1) <= a(n) <= n*a(n-1).
%e A226443 The sequence (i, j, k) has shadow transform (0, 1, m) where m is the number of even numbers in {i, j}, so a(3) = 3.
%o A226443 (PARI) sh(v)=vector(#v,i,my(n=i-1);sum(j=1,n,v[j]%n==0));
%o A226443 a(n)={
%o A226443   my(L=log(n+.5), t=primes(primepi(n)), D=divisors(prod(i=1,#t, t[i]^(L\log(t[i])))), nd=#D, v=[]);
%o A226443     for(i=1,nd^(n-1),
%o A226443         my(s=sh(vector(n,j,D[i\nd^(j-1)%nd+1])));
%o A226443         if(!setsearch(v,s),
%o A226443             v=vecsort(concat(v,[s]))
%o A226443         )
%o A226443     );
%o A226443     #v
%o A226443 };
%o A226443 (PARI) v=[]; fordiv(72,a, fordiv(72,b, fordiv(72,c, fordiv(72,d, fordiv(72,e, fordiv(72,f, fordiv(72,g, fordiv(72,h, fordiv(9,i, u=sh([a,b,c,d,e,f,g,h,i,0]); if(!vecsearch(v,u), v=vecsort(concat(v,[u])))))))))))); (5+1)*(7+1)*#v \\ computes a(10)
%Y A226443 Cf. A000522, A056793.
%K A226443 nonn,hard,more
%O A226443 0,4
%A A226443 _Charles R Greathouse IV_, Jun 06 2013