This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A118789 #14 Sep 01 2025 04:56:33
%S A118789 1,2,9,71,800,11659,208173,4398148,107293711,2967800711,91777098006,
%T A118789 3137581240925,117499040544197,4783424590188490,210333509575901445,
%U A118789 9934472399437068811,501615620424564184408,26963169913347131361647
%N A118789 Row sums of triangle A118788.
%C A118789 A032188 equals the main diagonal of triangle A118788; A032188(n) = number of labeled series-reduced mobiles (circular rooted trees) with n leaves.
%H A118789 Vaclav Kotesovec, <a href="/A118789/b118789.txt">Table of n, a(n) for n = 0..199</a>
%F A118789 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).
%F A118789 a(n) ~ n^n / (2 * exp(n - 1/2) * (1 - log(2))^(n + 1/2)). - _Vaclav Kotesovec_, Sep 01 2025
%e A118789 E.g.f.: A(x) = 1 + 1*x + 2*x^2/2! + 9*x^3/3! + 71*x^4/4! + ... =
%e A118789 exp(x + x^2/2! + 5*x^3/3! + 41*x^4/4! +... + A032188(n)*x^n/n! +...).
%t A118789 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 *)
%o A118789 (PARI)
%o A118789 {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))}
%Y A118789 Cf. A118788, A118790, A032188.
%K A118789 nonn,changed
%O A118789 0,2
%A A118789 _Paul D. Hanna_, Apr 29 2006