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

0,2

Comments

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

Links

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

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

Example

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! +...).

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 *)

Prog

(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))}
(PARI) \\ using function inverse_bell_matrix_row from A354794
a(n) = vecsum(inverse_bell_matrix_row(n+1, x->-x!)) \\ Mikhail Kurkov, May 12 2026

Crossrefs

Cf. A118788, A118790, A032188.

Sequence in context: A193469 A336606 A121879 * A386452 A393936 A258114

Adjacent sequences: A118786 A118787 A118788 * A118790 A118791 A118792

Keyword

nonn

Author

Paul D. Hanna, Apr 29 2006

Status

approved