|
%I #16 Nov 21 2022 09:10:07
%S 1,0,1,0,2,1,4,4,12,15,34,55,110,190,370,664,1272,2350,4466,8372,
%T 15926,30105,57390,109202,208738,398985,764906,1467370,2820770,
%U 5427543,10459456,20176561,38969684,75339232,145804978,282429242,547573768,1062501151,2063317650
%N Inverse Euler transform of A000016.
%D P. J. Cameron, Some counting problems related to permutation groups, Discrete Math., 225 (2000), 77-92.
%H Alois P. Heinz, <a href="/A057772/b057772.txt">Table of n, a(n) for n = 1..1000</a>
%p 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
%p # The function EulerInvTransform is defined in A358451.
%p a := EulerInvTransform(A000016):
%p seq(a(n), n = 1..39); # _Peter Luschny_, Nov 21 2022
%t 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_ *)
%K nonn
%O 1,5
%A _N. J. A. Sloane_, Nov 02 2000
%E Better definition and more terms from _Vladeta Jovovic_, Mar 13 2008
|
