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A056045 a(n) = Sum_{k|n} binomial(n,k). 22
1, 3, 4, 11, 6, 42, 8, 107, 94, 308, 12, 1718, 14, 3538, 3474, 14827, 18, 68172, 20, 205316, 117632, 705686, 24, 3587174, 53156, 10400952, 4689778, 41321522, 30, 185903342, 32, 611635179, 193542210, 2333606816, 7049188, 10422970784, 38 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..3329 (terms 1..500 from T. D. Noe)

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

FORMULA

L.g.f.: A(x) = Sum_{n>=1} log( G(x^n,n) ) where G(x,n) = 1 + x*G(x,n)^n. L.g.f. A(x) satisfies: exp(A(x)) = g.f. of A110448. - Paul D. Hanna, Nov 10 2007

a(n) = Sum_{k=1..A000005(n)} A007318(n, A027750(k)). - Reinhard Zumkeller, Aug 13 2013

EXAMPLE

A(x) = log(1/(1-x) * G(x^2,2) * G(x^3,3) * G(x^4,4) * ...)

where the functions G(x,n) are g.f.s of well-known sequences:

G(x,2) = g.f. of A000108 = 1 + x*G(x,2)^2;

G(x,3) = g.f. of A001764 = 1 + x*G(x,3)^3;

G(x,4) = g.f. of A002293 = 1 + x*G(x,4)^4; etc.

MATHEMATICA

f[n_] := Sum[ Binomial[n, d], {d, Divisors@ n}]; Array[f, 37] (* Robert G. Wilson v, Apr 23 2005 *)

Total[Binomial[#, Divisors[#]]]&/@Range[40] (* Harvey P. Dale, Dec 08 2018 *)

PROG

(PARI) {a(n)=n*polcoeff(sum(m=1, n, log(1/x*serreverse(x/(1+x^m +x*O(x^n))))), n)} /* Paul D. Hanna, Nov 10 2007 */

(PARI) {a(n)=sumdiv(n, d, binomial(n, d))} /* Paul D. Hanna, Nov 10 2007 */

(Haskell)

a056045 n = sum $ map (a007318 n) $ a027750_row n

-- Reinhard Zumkeller, Aug 13 2013

CROSSREFS

Cf. A110448 (exp(A(x)); A000108 (Catalan numbers), A001764, A002293, A174462.

Sequence in context: A197953 A198299 A175317 * A220848 A232891 A096223

Adjacent sequences:  A056042 A056043 A056044 * A056046 A056047 A056048

KEYWORD

nice,nonn

AUTHOR

Labos Elemer, Jul 25 2000

STATUS

approved

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Last modified October 14 17:43 EDT 2019. Contains 328022 sequences. (Running on oeis4.)