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L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} (1 + a(n)*x)^n * x^n/n.
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%I #4 Mar 07 2014 04:06:51

%S 1,3,10,59,796,38106,10575020,37219912979,4683360721197196,

%T 107669805691203995115748,4936018245619051863546606625582972,

%U 12131323997867394119748184355028213021384527189930

%N L.g.f.: Sum_{n>=1} a(n)*x^n/n = Sum_{n>=1} (1 + a(n)*x)^n * x^n/n.

%F a(n) = 1 + n*Sum_{k=1..[n/2]} C(n-k,k)*a(n-k)^k/(n-k) for n>1 with a(1)=1.

%e L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 59*x^4/4 + 796*x^5/5 +...

%e L(x) = (1+x)*x + (1+3*x)^2*x^2/2 + (1+10*x)^3*x^3/3 + (1+59*x)^4*x^4/4 +...

%e exp(L(x)) = g.f. of A158872 is an integer series:

%e exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 20*x^4 + 182*x^5 + 6552*x^6 +...

%t nmax = 15; a = ConstantArray[0, nmax]; a[[1]] = 1; Do[a[[n]] = 1 + n*Sum[Binomial[n-k,k]/(n-k) * a[[n-k]]^k, {k, 1, Floor[n/2]}], {n, 2, nmax}]; a (* _Vaclav Kotesovec_, Mar 07 2014 *)

%o (PARI) {a(n)=1+n*sum(k=1,n\2,binomial(n-k,k)*a(n-k)^k/(n-k))}

%Y Cf. A158872 (exp).

%K nonn

%O 1,2

%A _Paul D. Hanna_, Apr 10 2009