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A056045
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a(n) = Sum_{d|n} binomial(n,d).
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28
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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
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OFFSET
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1,2
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
(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
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CROSSREFS
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Cf. A000010 (comments on Dirichlet sum formulas).
Cf. A308943 (similar, with Product).
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KEYWORD
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nice,nonn
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AUTHOR
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STATUS
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approved
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