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
Keywords
Examples
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 +...] * ...
Crossrefs
Programs
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Mathematica
f[n_] := Block[{d = Divisors[n]}, n*Plus @@ (Binomial[n, d]/GCD[n, d])]; Table[ f[n], {n, 35}] -
PARI
a(n)=n*polcoeff(sum(m=1,n,m*log(1/x*serreverse(x/(1+x^m +x*O(x^n))))),n)
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PARI
a(n)=if(n<1,0,n*sumdiv(n,d,binomial(n,d)/gcd(n,d))) \\ Paul D. Hanna, Nov 11 2007
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
Comments