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A191461 E.g.f.: A(x) = Sum_{n>=0} binomial(x^n + n-1, n). 3

%I

%S 1,1,1,2,18,24,480,720,23520,100800,1874880,3628800,341953920,

%T 479001600,36739422720,468583315200,8960682931200,20922789888000,

%U 4289844445286400,6402373705728000,1012647189038284800,18895538930171904000,350378437544239104000

%N E.g.f.: A(x) = Sum_{n>=0} binomial(x^n + n-1, n).

%C The definition of the e.g.f. A(x) is an application of the identity

%C Sum_{n>=0} (-1)^n*log(1 - q^n*x)^n*y^n/n! = Sum_{n>=0} binomial(q^n*y +n-1, n)*x^n at x=1, y=1, q=x.

%F E.g.f.: A(x) = Sum_{n>=0} (-1)^n * log(1 - x^n)^n/n!.

%F E.g.f.: A(x) = Sum_{n>=0} Sum_{k=0..n} |Stirling1(n,k)|*x^(n*k)/n!.

%F a(p) = (p-1)! for prime p.

%e E.g.f.: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 18*x^4/4! + 24*x^5/5! +...

%e Series expansions:

%e A(x) = 1 - log(1 - x) + log(1 - x^2)^2/2! - log(1 - x^3)^3/3! + log(1 - x^4)^4/4! +...+ (-1)^n*log(1 - x^n)^n/n! +...

%e A(x) = 1 + x + x^2*(x^2+1)/2! + x^3*(x^3+1)*(x^3+2)/3! + x^4*(x^4+1)*(x^4+2)*(x^4+3)/4! +...+ binomial(x^n + n-1, n) +...

%e Coefficients a(n)/n! in the series expansion of the e.g.f. begin:

%e A(x) = 1 + 1/2*x + 1/3*x^2 + 3/4*x^3 + 1/5*x^4 + 2/3*x^5 + 1/7*x^6 + 7/12*x^7 + 5/18*x^8 + 31/60*x^9 + 1/11*x^10 + 257/360*x^11 + 1/13*x^12 +...

%e which illustrates the property a(p)/p! = 1/p for prime p.

%o (PARI) {a(n)=local(A=1+x); A=sum(m=0, n, binomial(x^m+n-1 +x*O(x^n), m)); n!*polcoeff(A, n)}

%o (PARI) {a(n)=local(A=1+x); A=1+sum(m=1, n, (-1)^m*log(1-x^m +x*O(x^n))^m/m!); n!*polcoeff(A, n)}

%o (PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}

%o {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, sum(k=0, m, abs(Stirling1(m, k))*x^(m*k))/m!)); n!*polcoeff(A, n)}

%Y Cf. A191460.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Jun 02 2011

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Last modified October 21 20:44 EDT 2019. Contains 328315 sequences. (Running on oeis4.)