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%I #18 May 24 2017 08:09:26
%S 1,2,14,164,2692,56832,1466656,44735392,1574507104,62807331520,
%T 2800211567936,137988945383808,7447519816062848,436913348544200192,
%U 27682538499602786816,1883880221782019929088,137046014280583363879936,10612885049611654523670528
%N E.g.f. A(x) satisfies: A( x - Sum_{n>=2} x^n/(n*(n-1)/2) ) = x.
%C Compare e.g.f. to the identity: let W(x) = Sum_{n>=1} (n-1)^(n-1)*x^n/n!, then W( x - Sum_{n>=2} x^n/(n*(n-1)) ) = x.
%F E.g.f. A(x) satisfies: A( -x - 2*(1-x)*log(1-x) ) = x.
%F a(n) = Sum_{k=1..n-1} A075856(n-1,k)*2^k for n>1 with a(1)=1.
%F a(n) ~ n^(n-1) / (sqrt(2) * (2-exp(1/2))^(n-1/2) * exp(n/2+1/2)). - _Vaclav Kotesovec_, Jan 23 2014
%F a(n) = (n-2)*a(n-1) + Sum_{j=1..n-1} binomial(n,j)*a(j)*a(n-j) for n>1, a(1)=1. - _Peter Luschny_, May 24 2017
%e E.g.f.: A(x) = x + 2*x^2/2! + 14*x^3/3! + 164*x^4/4! + 2692*x^5/5! + 56832*x^6/6! + 1466656*x^7/7! + 44735392*x^8/8! +...
%e Let R(x) be the series reversion of e.g.f. A(x), then R(x) begins:
%e R(x) = x - x^2/1 - x^3/3 - x^4/6 - x^5/10 - x^6/15 - x^7/21 - x^8/28 -...
%e ...
%e Compare to the series reversion of the function W(x) defined by:
%e W(x) = x + x^2/2! + 2^2*x^3/3! + 3^3*x^4/4! + 4^4*x^5/5! + 5^5*x^6/6! +...
%e where W(x - x^2/2 - x^3/6 - x^4/12 - x^5/20 - x^6/30 - x^7/42 -...) = x.
%p A209937_list := proc(len) local A, n; A[1] := 1; for n from 2 to len do
%p A[n] := (n-2)*A[n-1] + add(binomial(n,j)*A[j]*A[n-j], j=1..n-1) od:
%p convert(A,list) end: A209937_list(18); # _Peter Luschny_, May 24 2017
%t Rest[CoefficientList[InverseSeries[Series[-x-2*(1-x)*Log[1-x], {x, 0, 20}], x],x]*Range[0, 20]!] (* _Vaclav Kotesovec_, Jan 23 2014 *)
%o (PARI) {a(n)=n!*polcoeff(serreverse(x-sum(m=2, n, x^m/(m*(m-1)/2)) +x*O(x^n)), n)}
%o (PARI) {a(n)=n!*polcoeff(serreverse(-x-2*(1-x)*log(1-x +x*O(x^n))), n)}
%o for(n=1, 25, print1(a(n), ", "))
%o (PARI) /* From a formula of Michael Somos for triangle A075856: */
%o {A075856(n, k)=if(k<1|n<k,0,if(n==1,1,(n-1)*A075856(n-1,k)+(n+k-1)*A075856(n-1,k-1)))}
%o {a(n)=if(n<1,0,if(n==1,1,sum(k=1,n-1,A075856(n-1, k)*2^k)))}
%o for(n=1,20,print1(a(n),", "))
%Y Cf. A209923, A075856.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Mar 15 2012
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