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A209884 E.g.f. A(x) satisfies: A(x/(1-x))/(1-x) = (1/x) * d/dx x^2*A(x)/2.

%I A209884 #10 Nov 10 2016 03:50:06
%S A209884 1,2,10,88,1172,21688,527576,16224640,612742784,27786137856,
%T A209884 1486036616448,92365550416896,6591134271264000,534423153792751104,
%U A209884 48801114693060804096,4980221379342136676352,564172247490745614434304,70522894891787889987747840
%N A209884 E.g.f. A(x) satisfies: A(x/(1-x))/(1-x) = (1/x) * d/dx x^2*A(x)/2.
%F A209884 E.g.f.: exp( Sum_{n>=1} 2*x^n / (n*n!) ) = Sum_{n>=0} a(n)*x^n/n!^2.
%F A209884 a(n) = (n-1)! * Sum_{k=0..n-1} 2*binomial(n,k)*a(k)/k! for n>0 with a(0)=1.
%e A209884 E.g.f.: A(x) = 1 + 2*x + 10*x^2/2! + 88*x^3/3! + 1172*x^4/4! + 21688*x^5/5! +...
%e A209884 Related expansions:
%e A209884 A(x/(1-x))/(1-x) = 1 + 3*x + 20*x^2/2! + 220*x^3/3! + 3516*x^4/4! +...
%e A209884 A(x) + x*A'(x)/2 = 1 + 3*x + 20*x^2/2! + 220*x^3/3! + 3516*x^4/4! +...
%e A209884 Also, a(n) appears in the expansion:
%e A209884 B(x) = 1 + 2*x + 10*x^2/2!^2 + 88*x^3/3!^2 + 1172*x^4/4!^2 + 21688*x^5/5!^2 +...
%e A209884 such that
%e A209884 log(B(x)) = 2*x + 2*x^2/(2*2!) + 2*x^3/(3*3!) + 2*x^4/(4*4!) + 2*x^5/(5*5!) +...
%t A209884 Table[Sum[BellY[n, k, 2/Range[n]], {k, 0, n}] n!, {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)
%o A209884 (PARI) {a(n)=local(A=1+x, B); for(i=1, n, B=subst(A, x, x/(1-x+x*O(x^n)))/(1-x); A=1+2*intformal((B-A)/x)); n!*polcoeff(A, n)}
%o A209884 (PARI) {a(n)=if(n<0, 0, if(n==0, 1, (n-1)!*sum(k=0, n-1, 2*binomial(n, k)*a(k)/k!)))}
%o A209884 (PARI) {a(n)=n!^2*polcoeff(exp(sum(m=1, n, 2*x^m/(m*m!))+x*O(x^n)), n)}
%o A209884 for(n=0,30,print1(a(n),", "))
%Y A209884 Cf. A193161.
%K A209884 nonn
%O A209884 0,2
%A A209884 _Paul D. Hanna_, Mar 14 2012