A230321 - OEIS

%I #9 Jul 23 2014 01:54:17

%S 1,1,3,19,203,3296,75862,2340710,92647529,4554162028,271560907586,

%T 19291412245084,1608220567355569,155269058419296810,

%U 17162375811312467478,2150431928188151079196,302882085165757326494593,47608431893360236873620584,8298673711523249659301551906

%N E.g.f.: Sum_{n>=0} x^n/n! * Product_{k=1..n} LambertW(-k*x)/(-k*x).

%H Vaclav Kotesovec, <a href="/A230321/b230321.txt">Table of n, a(n) for n = 0..160</a>

%F E.g.f.: Sum_{n>=0} x^n/n! * Product_{k=1..n} Sum_{j>=0} k^j*(j+1)^(j-1)*x^j/j!.

%F E.g.f.: Sum_{n>=0} 1/n!^2 * Product_{k=1..n} -LambertW(-k*x).

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 137*x^4/4! + 1746*x^5/5! +...

%e Let W(x) = LambertW(-x)/(-x), then

%e W(k*x) = Sum_{j>=0} k^j*(j+1)^(j-1)*x^j/j!

%e where

%e A(x) = 1 + x*W(x) + x^2*W(x)*W(2*x)/2! + x^3*W(x)*W(2*x)*W(3*x)/3! + x^4*W(x)*W(2*x)*W(3*x)*W(4*x)/4! + x^5*W(x)*W(2*x)*W(3*x)*W(4*x)*W(5*x)/5! +...

%e RELATED EXPANSIONS:

%e W(1*x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! +...

%e W(2*x) = 1 + 2*x + 12*x^2/2! + 128*x^3/3! + 2000*x^4/4! + 41472*x^5/5! +...

%e W(3*x) = 1 + 3*x + 27*x^2/2! + 432*x^3/3! + 10125*x^4/4! + 314928*x^5/5! +...

%e W(4*x) = 1 + 4*x + 48*x^2/2! + 1024*x^3/3! + 32000*x^4/4! + 1327104*x^5/5! +...

%e W(5*x) = 1 + 5*x + 75*x^2/2! + 2000*x^3/3! + 78125*x^4/4! + 4050000*x^5/5! +...

%e ...

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

%o A=sum(m=0,n,x^m/m!*prod(k=1,m,subst(W,x,k*x)));

%o n!*polcoeff(A,n)}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) {a(n)=local(A=1);

%o A=sum(m=0,n,x^m/m!*prod(k=1,m,sum(j=0,n,k^j*(j+1)^(j-1)*x^j/j!)+x*O(x^n) ));

%o n!*polcoeff(A,n)}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) {a(n)=local(LambertW=serreverse(x*exp(x+x*O(x^n))),A=1);

%o A=sum(m=0,n,1/m!^2*prod(k=1,m,subst(-LambertW,x,-k*x)));

%o n!*polcoeff(A,n)}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A230320, A230317.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 15 2013