A108522 - OEIS

%I #20 Jan 20 2025 08:38:58

%S 1,2,9,70,771,10948,190205,3907494,92654059,2490459468,74827519077,

%T 2485153213814,90403692195179,3574835773247140,152675377606343901,

%U 7003761877546096278,343454890456254782203,17929588055863943650988

%N Number of increasing rooted trees with n generators.

%C A generator is a leaf or a node with just one child.

%C In an increasing rooted tree, nodes are numbered and numbers increase as you move away from root.

%H Vaclav Kotesovec, <a href="/A108522/b108522.txt">Table of n, a(n) for n = 1..250</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F E.g.f. satisfies: 2*A(x) = x - 1 + exp(A(x)) + Integral A(x) dx. - corrected by _Vaclav Kotesovec_ and _Paul D. Hanna_, Mar 29 2014

%F From _Paul D. Hanna_, Mar 29 2014: (Start)

%F E.g.f. satisfies: A(x) = A'(x)*(2 - exp(A(x))) - 1.

%F E.g.f. satisfies: A'(x) = (1 + A(x))/(2 - exp(A(x))).

%F (End)

%F a(n) ~ c * n^(n-1) / (exp(n) * r^n), where r = 0.3160173586544089316502903103262192204293322854083... and c = 0.51723490785798357350192800634304... - _Vaclav Kotesovec_, Mar 29 2014

%o (PARI) {a(n)=local(A=x);for(i=1,n,A=intformal((1+A)/(2-exp(A+x*O(x^n)))) );n!*polcoeff(A,n)}

%o for(n=1,20,print1(a(n),", ")) \\ _Paul D. Hanna_, Mar 29 2014

%Y Cf. A108521-A108529, A007151, A001147.

%K nonn,changed

%O 1,2

%A _Christian G. Bower_, Jun 07 2005