OFFSET
1,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..2503 (first 500 terms from Andrew Howroyd)
C. G. Bower, Transforms (2)
FORMULA
"CHK" (necklace, identity, unlabeled) transform of A004111.
MAPLE
g:= proc(n) option remember; `if`(n<2, n, add(g(n-k)*add(g(d)*d*
(-1)^(k/d+1), d=numtheory[divisors](k)), k=1..n-1)/(n-1))
end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(j-1-a(i), j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> g(n)+b(n, n-1):
seq(a(n), n=1..40); # Alois P. Heinz, May 19 2022
MATHEMATICA
g[n_] := g[n] = If[n < 2, n, Sum[g[n - k]*Sum[g[d]*d*(-1)^(k/d + 1), {d, Divisors[k]}], {k, 1, n - 1}]/(n - 1)];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[j - 1 - a[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := g[n] + b[n, n - 1];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, May 20 2022, after Alois P. Heinz *)
PROG
(PARI) \\ here IdTreeGf is g.f. of A004111.
IdTreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1) * d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
CHK(p, n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={Vec(CHK(IdTreeGf(n), n))} \\ Andrew Howroyd, Aug 31 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved