A056045 a(n) = Sum_{d|n} binomial(n,d).
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
Offset: 1
Examples
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.
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.
Crossrefs
Programs
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Haskell
a056045 n = sum $ map (a007318 n) $ a027750_row n -- Reinhard Zumkeller, Aug 13 2013
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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 *) -
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 */ -
Python
from math import comb from sympy import divisors def A056045(n): return sum(comb(n,d) for d in divisors(n,generator=True)) # Chai Wah Wu, Jul 22 2024
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..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