A206763 G.f.: Product_{n>=1} [ (1 - (-x)^n) / (1 - (n-1)^n*x^n) ]^(1/n).
1, 1, 0, 3, 23, 225, 2824, 42670, 762286, 15647321, 363901749, 9443387329, 270721307582, 8493470965716, 289518611494068, 10653599202688527, 420933469388468297, 17773313165985120798, 798686060913371460133, 38058408270727983373232
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 3*x^3 + 23*x^4 + 225*x^5 + 2824*x^6 + 42670*x^7 +... where the g.f. equals the product: A(x) = (1+x)/(1-0*x) * ((1-x^2)/(1-1^2*x^2))^(1/2) * ((1+x^3)/(1-2^3*x^3))^(1/3) * ((1-x^4)/(1-3^4*x^4))^(1/4) * ((1+x^5)/(1-4^5*x^5))^(1/5) *... The logarithm equals the l.g.f. of A206764: log(A(x)) = x - x^2/2 + 10*x^3/3 + 79*x^4/4 + 1026*x^5/5 + 15686*x^6/6 +...
Programs
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PARI
{a(n)=polcoeff(prod(k=1, n, ((1-(-1)^k*x^k)/(1-(k-1)^k*x^k +x*O(x^n)))^(1/k)), n)} -
PARI
{a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*sum(k=1, m, binomial(m, k)*sigma(m, k)*(-1)^(m-k))+x*O(x^n))), n)} for(n=0,31,print1(a(n),", "))
Formula
G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=1..n} binomial(n,k) * sigma(n,k) * (-1)^(n-k) ).
Logarithmic derivative yields A206764.
Comments