%I #17 May 02 2020 16:40:12
%S 1,1,3,12,52,247,1226,6299,33209,178618,976296,5407384,30283120,
%T 171196956,975662480,5599508648,32334837886,187737500013,
%U 1095295264857,6417886638389,37752602033079,222861754454841,1319834477009635,7839314017612273,46688045740233741
%N Number of rooted identity trees with 2n+1 nodes.
%H Alois P. Heinz, <a href="/A299113/b299113.txt">Table of n, a(n) for n = 0..1253</a>
%F a(n) = A004111(2n+1).
%e a(2) = 3:
%e o o o
%e | | / \
%e o o o o
%e | / \ |
%e o o o o
%e | | |
%e o o o
%e |
%e o
%p with(numtheory):
%p b:= proc(n) option remember; `if`(n<2, n, add(b(n-k)*add(
%p b(d)*d*(-1)^(k/d+1), d=divisors(k)), k=1..n-1)/(n-1))
%p end:
%p a:= n-> b(2*n+1):
%p seq(a(n), n=0..30);
%t b[n_] := b[n] = If[n < 2, n, Sum[b[n - k]*Sum[b[d]*d*(-1)^(k/d + 1), {d, Divisors[k]}], {k, 1, n - 1}]/(n - 1)];
%t a[n_] := b[2*n + 1];
%t Array[a, 30, 0] (* _Jean-François Alcover_, May 30 2019, from Maple *)
%o (Python)
%o from sympy import divisors
%o from sympy.core.cache import cacheit
%o @cacheit
%o def b(n): return n if n<2 else sum([b(n-k)*sum([b(d)*d*(-1)**(k//d+1) for d in divisors(k)]) for k in range(1, n)])//(n-1)
%o def a(n): return b(2*n+1)
%o print([a(n) for n in range(31)]) # _Indranil Ghosh_, Mar 02 2018
%Y Bisection of A004111 (odd part).
%Y Cf. A100427, A299098.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Feb 02 2018