A278332 E.g.f. equals the limit of the average of all permutations of the compositions of the functions x*exp(x^k), for k=1..n, as n increases.
1, 2, 9, 88, 905, 12666, 220297, 4506440, 104707521, 2758506850
Offset: 1
Examples
E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 88*x^4/4! + 905*x^5/5! + 12666*x^6/6! + 220297*x^7/7! + 4506440*x^8/8! + 104707521*x^9/9! + 2758506850*x^10/10! +... Generating method. Define F(n,x) as the average of the sum over all n! permutations of the compositions of x*exp(x^k) for k=1..n, then the e.g.f. of this sequence is the limit of the functions F(n,x) as n grows. Examples of some initial functions F(n,x) are as follows. At n=1, F(1,x) = x*exp(x). At n=2, F(2,x) = (1/2!)*([x*exp(x) o x*exp(x^2)] + [x*exp(x^2) o x*exp(x)]) = x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 425*x^5/5! + 3486*x^6/6! +... At n=3, F(3,x) = (1/3!)*([x*exp(x) o x*exp(x^2) o x*exp(x^3)] + [x*exp(x^2) o x*exp(x) o x*exp(x^3)] + [x*exp(x) o x*exp(x^3) o x*exp(x^2)] + [x*exp(x^3) o x*exp(x) o x*exp(x^2)] + [x*exp(x^2) o x*exp(x^3) o x*exp(x)] + [x*exp(x^3) o x*exp(x^2) o x*exp(x)]) = x + 2*x^2/2! + 9*x^3/3! + 88*x^4/4! + 785*x^5/5! + 9426*x^6/6! +... etc.
Programs
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PARI
/* Informal code to generate terms (ran over 100 hrs for 10 terms!) */ {F(n) = x*exp(x^n +x*O(x^16))} {P(n) = vector(n!,k, numtoperm(n,k) )} {A(n) = my(H=0); for(k=1,n!, G=x; for(j=1,n, G=subst(G,x, F(P(n)[k][j]) )); H=H+G);H/n!} {Vec(serlaplace(A(6)))}
Formula
E.g.f.: A(x) = limit_{n->oo} (1/n!) * Sum(all permutations of compositions of functions x*exp(x^k) for k=1..n).