login
Number of linear forests with n rooted trees and 2*n-1 nodes.
3

%I #15 Jan 03 2021 06:40:13

%S 0,1,2,9,44,230,1236,6790,37832,213057,1209660,6912367,39705516,

%T 229055918,1326168018,7701734250,44846271632,261735599172,

%U 1530650010312,8967361033572,52619233554120,309203221308702,1819290987055630,10716835948503349,63196331969007264

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

%H Alois P. Heinz, <a href="/A339440/b339440.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = A339067(2n-1,n).

%F a(n) ~ c * d^n / sqrt(n), where d = 6.031382795097860532993547039674008662345079835351392549515262162478014679... and c = 0.05599525103242350197279211300654208236718263537075... - _Vaclav Kotesovec_, Dec 18 2020

%p b:= proc(n) option remember; `if`(n<2, n, (add(add(d*b(d),

%p d=numtheory[divisors](j))*b(n-j), j=1..n-1))/(n-1))

%p end:

%p T:= proc(n, k) option remember; `if`(k=1, b(n), (t->

%p add(T(j, t)*T(n-j, k-t), j=1..n-1))(iquo(k, 2)))

%p end:

%p a:= n-> T(2*n-1, n):

%p seq(a(n), n=0..24);

%t 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 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 a[n_] := T[2n-1, n];

%t a /@ Range[0, 24] (* _Jean-François Alcover_, Jan 03 2021, after _Alois P. Heinz_ *)

%Y Cf. A339067, A245102, A245103.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Dec 04 2020