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%I #20 Jun 25 2021 03:40:14
%S 1,3,21,249,4356,103932,3213216,124146432,5834291328,326570493312,
%T 21408981213888,1621281984076224,140205279698051328,
%U 13711076231477352192,1503606581609959001088,183562416179374733411328,24787906630769478567297024
%N E.g.f. A(x) satisfies: A(x/(1-x))/(1-x) = (1/x^2) * d/dx x^3*A(x)/3.
%H Vincenzo Librandi, <a href="/A209917/b209917.txt">Table of n, a(n) for n = 0..100</a>
%F E.g.f.: exp( Sum_{n>=1} 3*x^n/(n*n!) ) = Sum_{n>=0} a(n)*x^n/n!^2.
%F a(n) = (n-1)!* Sum_{k=0..n-1} 3*binomial(n,k)*a(k)/k! for n>0 with a(0)=1.
%e E.g.f.: A(x) = 1 + 3*x + 21*x^2/2! + 249*x^3/3! + 4356*x^4/4! + 103932*x^5/5! + ...
%e Related expansions:
%e A(x/(1-x))/(1-x) = 1 + 4*x + 35*x^2/2! + 498*x^3/3! + 10164*x^4/4! + ...
%e A(x) + x*A'(x)/3 = 1 + 4*x + 35*x^2/2! + 498*x^3/3! + 10164*x^4/4! + ...
%e Also, a(n) appears in the expansion:
%e B(x) = 1 + 3*x + 21*x^2/2!^2 + 249*x^3/3!^2 + 4356*x^4/4!^2 + 103932*x^5/5!^2 + ...
%e such that
%e log(B(x)) = 3*x + 3*x^2/(2*2!) + 3*x^3/(3*3!) + 3*x^4/(4*4!) + 3*x^5/(5*5!) + ...
%t a = ConstantArray[0,21]; a[[1]]=1; a[[2]]=3; Do[a[[n+1]] = (n-1)!*Sum[3*Binomial[n, k]*a[[k+1]]/k!,{k,0,n-1}],{n,2,20}]; a (* _Vaclav Kotesovec_, Feb 23 2014 *)
%t Table[Sum[BellY[n, k, 3/Range[n]], {k, 0, n}] n!, {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)
%o (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+3*intformal((B-A)/x)); n!*polcoeff(A, n)}
%o (PARI) {a(n)=if(n<0, 0, if(n==0, 1, (n-1)!*sum(k=0, n-1, 3*binomial(n, k)*a(k)/k!)))}
%o (PARI) {a(n)=n!^2*polcoeff(exp(sum(m=1, n, 3*x^m/(m*m!))+x*O(x^n)), n)}
%o for(n=0,30,print1(a(n),", "))
%o (Sage)
%o @CachedFunction
%o def a(n): return 1 if (n==0) else factorial(n-1)*sum( 3*binomial(n, j)*a(j)/factorial(j) for j in (0..n-1) )
%o [a(n) for n in (0..20)] # _G. C. Greubel_, Jun 23 2021
%Y Cf. A193161, A209884.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Mar 15 2012
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