|
%I #5 Jun 11 2014 00:44:04
%S 1,2,6,24,123,786,6112,56504,607833,7467066,103120674,1580416008,
%T 26598861595,487397118314,9654657563748,205508121922824,
%U 4676833854264603,113293413849825702,2910255267932697130,79008535866112062440,2260164362082172615833,67947314033110789074486
%N G.f.: exp( Integral Sum_{n>=1} (n+1)!*x^(n-1) / Product_{k=1..n} (1-k*x) dx ).
%C Compare g.f. to:
%C exp( Integral Sum_{n>=1} (n+1)!*x^(n-1) / Product_{k=1..n} (1+k*x) dx ) = 1/(1-2*x).
%F G.f.: exp( Sum_{n>=1} A005649(n)*x^n/n ) where Sum_{n>=0} A005649(n)*x^n/n! = 1/(2-exp(x))^2.
%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 24*x^3 + 123*x^4 + 786*x^5 + 6112*x^6 +...
%e where the logarithmic derivative is given by the series:
%e A'(x)/A(x) = 2!/(1-x) + 3!*x/((1-x)*(1-2*x)) + 4!*x^2/((1-x)*(1-2*x)*(1-3*x)) + 5!*x^3/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)) + 6!*x^4/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)) +...
%e Explicitly,
%e A'(x)/A(x) = 2 + 8*x + 44*x^2 + 308*x^3 + 2612*x^4 + 25988*x^5 + 296564*x^6 +...+ A005649(n+1)*x^n +...
%e compare to:
%e 1/(2-exp(x))^2 = 1 + 2*x + 8*x^2/2! + 44*x^3/3! + 308*x^4/4! + 2612*x^5/5! + 25988*x^6/6! +...+ A005649(n)*x^n/n! +...
%o (PARI) {a(n)=local(A=1+x);A=exp(intformal(sum(m=1,n+1,(m+1)!*x^(m-1)/prod(k=1,m,1-k*x +x*O(x^n)))));polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) /* From g.f. exp( Sum_{n>=1} A005649(n)*x^n/n ): */
%o {A005649(n)=n!*polcoeff(1/(2-exp(x +x*O(x^n)))^2,n)}
%o {a(n)=polcoeff(exp(sum(m=1,n,A005649(n)*x^n/n) +x*O(x^n)),n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A005649.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 11 2014
| 