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

1,2

Links

Table of n, a(n) for n=1..10.

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).

Example

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.

Prog

(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)))}

Crossrefs

Cf. A278333(n) = a(n+1)/(n+1) for n>=0.

Cf. A277180, A277181.

Sequence in context: A368186 A359903 A360433 * A135747 A270862 A259794

Adjacent sequences: A278329 A278330 A278331 * A278333 A278334 A278335

Keyword

nonn,more

Author

Paul D. Hanna, Nov 18 2016

Status

approved