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%I #13 Nov 01 2014 04:57:03
%S 1,1,3,16,137,1746,31627,785149,25715377,1070214364,54862242971,
%T 3385895548839,247409460018217,21118696317592498,2080845352648353215,
%U 234093630772343822281,29777361783749418754593,4247066958924682143019576,674393753569770072828136819
%N E.g.f.: Sum_{n>=0} x^n/n! * Product_{k=1..n} ( LambertW(-k*x)/(-k*x) )^(1/k).
%H Vaclav Kotesovec, <a href="/A230320/b230320.txt">Table of n, a(n) for n = 0..150</a>
%F E.g.f.: Sum_{n>=0} x^n/n! * Product_{k=1..n} Sum_{j>=0} (k*j+1)^(j-1)*x^j/j!.
%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)^(1/k) = Sum_{j>=0} (k*j+1)^(j-1)*x^j/j!
%e where
%e A(x) = 1 + x*W(x) + x^2*W(x)*W(2*x)^(1/2)/2! + x^3*W(x)*W(2*x)^(1/2)*W(3*x)^(1/3)/3! + x^4*W(x)*W(2*x)^(1/2)*W(3*x)^(1/3)*W(4*x)^(1/4)/4! +...
%e RELATED EXPANSIONS:
%e W(1*x)^(1/1) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! +...
%e W(2*x)^(1/2) = 1 + x + 5*x^2/2! + 49*x^3/3! + 729*x^4/4! + 14641*x^5/5! +...
%e W(3*x)^(1/3) = 1 + x + 7*x^2/2! + 100*x^3/3! + 2197*x^4/4! + 65536*x^5/5! +...
%e W(4*x)^(1/4) = 1 + x + 9*x^2/2! + 169*x^3/3! + 4913*x^4/4! + 194481*x^5/5! +...
%e W(5*x)^(1/5) = 1 + x + 11*x^2/2! + 256*x^3/3! + 9261*x^4/4! + 456976*x^5/5! +...
%e W(6*x)^(1/6) = 1 + x + 13*x^2/2! + 361*x^3/3! + 15625*x^4/4! + 923521*x^5/5! +...
%e W(7*x)^(1/7) = 1 + x + 15*x^2/2! + 484*x^3/3! + 24389*x^4/4! + 1679616*x^5/5! +...
%e W(8*x)^(1/8) = 1 + x + 17*x^2/2! + 625*x^3/3! + 35937*x^4/4! + 2825761*x^5/5! +...
%e W(9*x)^(1/9) = 1 + x + 19*x^2/2! + 784*x^3/3! + 50653*x^4/4! + 4477456*x^5/5! +...
%e W(10*x)^(1/10) = 1 + x + 21*x^2/2! + 961*x^3/3! + 68921*x^4/4! + 6765201*x^5/5! +...
%o (PARI) {a(n)=local(LambertW=serreverse(x*exp(x+x^2*O(x^n))), A=1);
%o A=sum(m=0, n, x^m/m!*prod(k=1, m, (subst(LambertW, x, -k*x)/(-k*x))^(1/k)));
%o n!*polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%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)^(1/k)));
%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+1)^(j-1)*x^j/j!)+x*O(x^n) ));
%o n!*polcoeff(A,n)}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A230321, A230318.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 15 2013
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