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A339440 Number of linear forests with n rooted trees and 2*n-1 nodes.

%I A339440 #15 Jan 03 2021 06:40:13
%S A339440 0,1,2,9,44,230,1236,6790,37832,213057,1209660,6912367,39705516,
%T A339440 229055918,1326168018,7701734250,44846271632,261735599172,
%U A339440 1530650010312,8967361033572,52619233554120,309203221308702,1819290987055630,10716835948503349,63196331969007264
%N A339440 Number of linear forests with n rooted trees and 2*n-1 nodes.
%H A339440 Alois P. Heinz, <a href="/A339440/b339440.txt">Table of n, a(n) for n = 0..1000</a>
%F A339440 a(n) = A339067(2n-1,n).
%F A339440 a(n) ~ c * d^n / sqrt(n), where d = 6.031382795097860532993547039674008662345079835351392549515262162478014679... and c = 0.05599525103242350197279211300654208236718263537075... - _Vaclav Kotesovec_, Dec 18 2020
%p A339440 b:= proc(n) option remember; `if`(n<2, n, (add(add(d*b(d),
%p A339440       d=numtheory[divisors](j))*b(n-j), j=1..n-1))/(n-1))
%p A339440     end:
%p A339440 T:= proc(n, k) option remember; `if`(k=1, b(n), (t->
%p A339440       add(T(j, t)*T(n-j, k-t), j=1..n-1))(iquo(k, 2)))
%p A339440     end:
%p A339440 a:= n-> T(2*n-1, n):
%p A339440 seq(a(n), n=0..24);
%t A339440 b[n_] := b[n] = If[n<2, n, (Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n - j], {j, 1, n - 1}])/(n - 1)];
%t A339440 T[n_, k_] := T[n, k] = If[k == 1, b[n], With[{t = Quotient[k, 2]}, Sum[T[j, t]*T[n - j, k - t], {j, 1, n - 1}]]];
%t A339440 a[n_] := T[2n-1, n];
%t A339440 a /@ Range[0, 24] (* _Jean-François Alcover_, Jan 03 2021, after _Alois P. Heinz_ *)
%Y A339440 Cf. A339067, A245102, A245103.
%K A339440 nonn
%O A339440 0,3
%A A339440 _Alois P. Heinz_, Dec 04 2020