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A056045 a(n) = Sum_{d|n} binomial(n,d). 27
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
a(n) = Sum_{k=1..n} binomial(n,gcd(n,k))/phi(n/gcd(n,k)) = Sum_{k=1..n} binomial(n,n/gcd(n,k))/phi(n/gcd(n,k)) where phi = A000010. - Richard L. Ollerton, Nov 08 2021
a(n) = n+1 iff n is prime. - Bernard Schott, Nov 30 2021
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.
Cf. A000010 (comments on Dirichlet sum formulas).
Cf. A308943 (similar, with Product).
Sequence in context: A198299 A360948 A175317 * A360794 A220848 A370605
KEYWORD
nice,nonn
AUTHOR
Labos Elemer, Jul 25 2000
STATUS
approved

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Last modified May 29 00:29 EDT 2024. Contains 372921 sequences. (Running on oeis4.)