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A055891 CIK (necklace, indistinct, unlabeled) transform of powers of 2. 1
1, 2, 7, 20, 64, 200, 686, 2324, 8194, 29084, 104860, 381116, 1398148, 5161592, 19173958, 71580752, 268435474, 1010572832, 3817749138, 14467230668, 54975581488, 209430687944, 799644820114, 3059510251700, 11728124035248 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

From Petros Hadjicostas, Dec 06 2017: (Start)

The g.f. is clear from J. Arndt's PARI program below.

The CIK transform of sequence (a(n): n>=1} with g.f. A(x) = Sum_{n>=1} a(n)*x^n has g.f. CIK(A(x)) = 1 - Sum_{n>=1} (phi(n)/n)*log(1-A(x^n)). Sometimes, the constant 1 is dropped from the formula. Here, A(x) = 2*x/(1-2*x).

To find the auxiliary sequence (c(n): n>=1} used in the formula a(n) = (1/n)*Sum_{d|n} phi(n/d)*c(d), we use the formula C(x) = Sum_{n>=1} c(n)*x^n = x*(dA(x)/dx)/(1-A(x)). Here, C(x) = 2*x/((1-4*x)*(1-2*x)), from which we can prove that c(n) = 2^n*(2^n-1) = A020522(n).

(End)

LINKS

Table of n, a(n) for n=0..24.

C. G. Bower, Transforms (2)

P. Flajolet and M. Soria, The Cycle Construction, SIAM J. Discr. Math., vol. 4 (1), 1991, pp. 58-60.

FORMULA

From Petros Hadjicostas, Dec 06 2017: (Start)

a(n) = (1/n)*Sum_{d|n} phi(n/d)*2^d*(2^d-1) = (1/n)*Sum_{d|n} phi(n/d)*A020522(d) for n >= 1.

G.f.: 1 - Sum_{n>=1} (phi(n)/n)*log((1-4*x^n)/(1-2*x^n)).

(End)

MATHEMATICA

{1}~Join~Table[(1/n) DivisorSum[n, EulerPhi[n/#] *2^#*(2^# - 1) &], {n, 24}] (* Michael De Vlieger, Dec 06 2017 *)

PROG

(PARI)

N = 66;  x = 'x + O('x^N);

f(x)=sum(n=1, N, 2^n*x^n );

gf = 1 + sum(n=1, N, eulerphi(n)/n*log(1/(1-f(x^n)))  );

v = Vec(gf)

/* Joerg Arndt, Jan 21 2013 */

CROSSREFS

Cf. A000079, A020522, A055376.

Sequence in context: A216246 A000935 A035071 * A122877 A192680 A000150

Adjacent sequences:  A055888 A055889 A055890 * A055892 A055893 A055894

KEYWORD

nonn

AUTHOR

Christian G. Bower, Jun 09 2000

STATUS

approved

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Last modified October 19 12:07 EDT 2018. Contains 316360 sequences. (Running on oeis4.)