A057772 Inverse Euler transform of A000016.

1, 0, 1, 0, 2, 1, 4, 4, 12, 15, 34, 55, 110, 190, 370, 664, 1272, 2350, 4466, 8372, 15926, 30105, 57390, 109202, 208738, 398985, 764906, 1467370, 2820770, 5427543, 10459456, 20176561, 38969684, 75339232, 145804978, 282429242, 547573768, 1062501151, 2063317650

Offset: 1

Author

N. J. A. Sloane, Nov 02 2000

Keywords

• nonn

References

• P. J. Cameron, Some counting problems related to permutation groups, Discrete Math., 225 (2000), 77-92.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Programs

Maple

with(numtheory): ietr:= proc(p) local a, c; c:= proc(n) option remember; local j; n*p(n)-add(c(j)*p(n-j), j=1..n-1) end; a:=proc(n) option remember; local d; `if`(n=0,1, add(mobius(n/d)*c(d), d=divisors(n))/n) end end: a:= ietr(n-> add(phi(d) *2^(n/d)/2/n, d=select(m-> modp(m,2)=1, divisors(n)))): seq(a(n), n=1..40); # Alois P. Heinz, Sep 08 2008
# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(A000016):
seq(a(n), n = 1..39); # Peter Luschny, Nov 21 2022

Mathematica

ietr[p_] := Module[{a, c}, c[n_] := c[n] = Module[{j}, n*p[n] - Sum[c[j]*p[n-j], {j, 1, n-1}]]; a[n_] := a[n] = Module[{d}, If[n == 0, 1, Sum[MoebiusMu[n/d]*c[d], {d, Divisors[n]}]/n]]; a]; a = ietr[Function[n, Sum[EulerPhi[d]*2^(n/d)/2/n, {d, Select[Divisors[n], OddQ]}]]]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jan 17 2014, after Alois P. Heinz *)

Extensions

Better definition and more terms from Vladeta Jovovic, Mar 13 2008