The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 Adjacent sequences: A056042 A056043 A056044 * A056046 A056047 A056048 KEYWORD nice,nonn AUTHOR Labos Elemer, Jul 25 2000 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

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