A105862 a(n) = n * Sum_{d|n} binomial(n,d)/gcd(n,d).

1, 5, 10, 29, 26, 122, 50, 317, 334, 830, 122, 4754, 170, 7698, 11510, 34237, 290, 159530, 362, 458054, 358592, 1413890, 530, 8236946, 266276, 20806102, 14087530, 85118762, 842, 404242022, 962, 1244530621, 580671266, 4667223134, 35896250

Offset

1,2

Links

Table of n, a(n) for n=1..35.

Formula

a(n) = n * sum_{d|n} (binomial(n, d) / GCD(n, d)).

L.g.f.: A(x) = Sum_{n>=1} LOG[ G(x^n,n)^n ] where G(x,n) = 1 + x*G(x,n)^n, where exp(A(x)) = g.f. of A110448. - Paul D. Hanna, Nov 11 2007

Example

L.g.f.: A(x) = x + 5/2*x^2 + 10/3*x^3 + 29/4*x^4 + 26/5*x^5 + 61/3*x^6 +...
L.g.f.: A(x) = LOG[1/(1-x) * G(x^2,2)^2 * G(x^3,3)^3 * G(x^4,4)^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.
Exponentiation of l.g.f. A(x) is expressed by a product that begins:
exp(A(x)) = [1 + x + x^2 + x^3 +...] * [1 + 2*x^2 + 5*x^4 + 14*x^6 +...] * [1 + 3*x^3 + 12*x^6 + 55*x^9 +...] * [1 + 4*x^4 + 22*x^8 + 140*x^12 +...] * ...

Mathematica

f[n_] := Block[{d = Divisors[n]}, n*Plus @@ (Binomial[n, d]/GCD[n, d])]; Table[ f[n], {n, 35}]

Prog

(PARI) a(n)=n*polcoeff(sum(m=1, n, m*log(1/x*serreverse(x/(1+x^m +x*O(x^n))))), n)
(PARI) a(n)=if(n<1, 0, n*sumdiv(n, d, binomial(n, d)/gcd(n, d))) \\ Paul D. Hanna, Nov 11 2007

Crossrefs

Cf. A105861, A105863.

Cf. A134774 (exp(A(x)); A056045 (variant); A000108 (Catalan), A001764, A002293.

Sequence in context: A022094 A134129 A240515 * A338662 A093029 A105505

Adjacent sequences: A105859 A105860 A105861 * A105863 A105864 A105865

Keyword

nonn

Author

Robert G. Wilson v, Apr 23 2005

Status

approved