A118789 Row sums of triangle A118788.

1, 2, 9, 71, 800, 11659, 208173, 4398148, 107293711, 2967800711, 91777098006, 3137581240925, 117499040544197, 4783424590188490, 210333509575901445, 9934472399437068811, 501615620424564184408, 26963169913347131361647

Offset: 0

Author

Paul D. Hanna, Apr 29 2006

Keywords

• nonn
• changed

Comments

A032188 equals the main diagonal of triangle A118788; A032188(n) = number of labeled series-reduced mobiles (circular rooted trees) with n leaves.

Examples

			E.g.f.: A(x) = 1 + 1*x + 2*x^2/2! + 9*x^3/3! + 71*x^4/4! + ... =
exp(x + x^2/2! + 5*x^3/3! + 41*x^4/4! +... + A032188(n)*x^n/n! +...).
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..199

Crossrefs

Cf. A118788, A118790, A032188.

Programs

Mathematica

Table[Sum[n!/(n-k)! * SeriesCoefficient[(x/(2*x + Log[1-x]))^(n + 1), {x, 0, k}], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 01 2025 *)

PARI

{a(n)=local(x=X+X^2*O(X^n));sum(k=0,n,n!/(n-k)!*polcoeff((x/(2*x+log(1-x)))^(n+1),k,X))}

Formula

E.g.f.: A(x) = exp( Sum_{n>=1} A032188(n)*x^n/n! ). As row sums of A118788, a(n) = Sum_{k=0..n} n!/(n-k)!*[x^k]{ x/(2*x + log(1-x)) }^(n+1).

a(n) ~ n^n / (2 * exp(n - 1/2) * (1 - log(2))^(n + 1/2)). - Vaclav Kotesovec, Sep 01 2025