A088791 Coefficient of x^n in A(x)^n is A000670(n), which gives preferential arrangements of n labeled elements.

1, 1, 1, 2, 8, 46, 337, 2976, 30627, 359222, 4725968, 68903766, 1102712316, 19219507328, 362428546833, 7352854216056, 159705991698432, 3697928742242694, 90933523698184947, 2366758931071064064

Offset: 0

Author

Paul D. Hanna, Oct 16 2003

Keywords

• nonn
• easy

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..333

Crossrefs

Cf. A000670, A084784.

Programs

Mathematica

nmax = 19; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^2 - (A[x A[x]] + x A[x]) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 02 2019 *)

PARI

{a(n)=local(A, m); if(n<1, n==0, m=1; A=1+x; for(i=1, n, A=(subst(A, x, x*A+x*O(x^n)) + x*A)/A); polcoeff(A, n))}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Feb 11 2015

Formula

G.f. satisfies: A(x)^2 = A(x*A(x)) + x*A(x).

a(n) ~ (n-1)! / (4 * (log(2))^(n+1)). - Vaclav Kotesovec, Feb 12 2015

O.g.f.: A(x) = x/( series reversion x*B(x) ), where B(x) = 1 + x + 2*x^2 + 6*x^3 + 25*x^4 + 137*x^5 + ... is the o.g.f. of A084784. - Peter Bala, Jun 23 2015