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

0, 1, 2, 9, 44, 230, 1236, 6790, 37832, 213057, 1209660, 6912367, 39705516, 229055918, 1326168018, 7701734250, 44846271632, 261735599172, 1530650010312, 8967361033572, 52619233554120, 309203221308702, 1819290987055630, 10716835948503349, 63196331969007264

Offset: 0

Author

Alois P. Heinz, Dec 04 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A339067, A245102, A245103.

Programs

Maple

b:= proc(n) option remember; `if`(n<2, n, (add(add(d*b(d),
      d=numtheory[divisors](j))*b(n-j), j=1..n-1))/(n-1))
    end:
T:= proc(n, k) option remember; `if`(k=1, b(n), (t->
      add(T(j, t)*T(n-j, k-t), j=1..n-1))(iquo(k, 2)))
    end:
a:= n-> T(2*n-1, n):
seq(a(n), n=0..24);

Mathematica

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[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}]]];
a[n_] := T[2n-1, n];
a /@ Range[0, 24] (* Jean-François Alcover, Jan 03 2021, after Alois P. Heinz *)

Formula

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

a(n) ~ c * d^n / sqrt(n), where d = 6.031382795097860532993547039674008662345079835351392549515262162478014679... and c = 0.05599525103242350197279211300654208236718263537075... - Vaclav Kotesovec, Dec 18 2020