A000300 4th power of rooted tree enumerator: linear forests of 4 rooted trees.

1, 4, 14, 44, 133, 388, 1116, 3168, 8938, 25100, 70334, 196824, 550656, 1540832, 4314190, 12089368, 33911543, 95228760, 267727154, 753579420, 2123637318, 5991571428, 16923929406, 47857425416, 135478757308, 383929643780, 1089118243128, 3092612497260

Offset: 4

Author

N. J. A. Sloane

Keywords

• nonn

References

• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 4..500
• Index entries for sequences related to rooted trees

Crossrefs

Column 4 of A339067.

Cf. A000081, A000106, A000242, A000343, A000395.

Programs

Maple

b:= proc(n) option remember; if n<=1 then n else add(k*b(k)* s(n-1, k), k=1..n-1)/(n-1) fi end: s:= proc(n,k) option remember; add(b(n+1-j*k), j=1..iquo(n,k)) end: B:= proc(n) option remember; add(b(k)*x^k, k=1..n) end: a:= n-> coeff(series(B(n-3)^4, x=0, n+1), x,n): seq(a(n), n=4..30); # Alois P. Heinz, Aug 21 2008

Mathematica

b[n_] := b[n] = If[ n <= 1, n, Sum[k*b[k]*s[n-1, k], {k, 1, n-1}]/(n-1)]; s[n_, k_] := s[n, k] = Sum[ b[n + 1 - j*k], {j, 1, n/k}]; bb[n_] := bb[n] = Sum[b[k]*x^k, {k, 1, n}]; a[n_] := Coefficient[ Series[ bb[n - 3]^4, {x, 0, n + 1}], x, n]; Table[a[n], {n, 4, 31}] (* Jean-François Alcover, Jan 25 2013, translated from Alois P. Heinz's Maple program *)

Formula

G.f.: B(x)^4 where B(x) is g.f. of A000081.

a(n) ~ 4 * A187770 * A051491^n / n^(3/2). - Vaclav Kotesovec, Jan 03 2021

Extensions

More terms from Christian G. Bower, Nov 15 1999