A000300 4th power of rooted tree enumerator: linear forests of 4 rooted trees. (Formerly M3479 N1414)

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

4,2

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

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

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 *)

Crossrefs

Column 4 of A339067.

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

Sequence in context: A118042 A006645 A094309 * A005323 A027831 A097894

Adjacent sequences: A000297 A000298 A000299 * A000301 A000302 A000303

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Christian G. Bower, Nov 15 1999

Status

approved