A000235 Number of n-node rooted trees of height 3. (Formerly M2732 N1097)

0, 0, 0, 1, 3, 8, 18, 38, 76, 147, 277, 509, 924, 1648, 2912, 5088, 8823, 15170, 25935, 44042, 74427, 125112, 209411, 348960, 579326, 958077, 1579098, 2593903, 4247768, 6935070, 11290627, 18330973, 29684082, 47946852, 77258764, 124198083

Offset

1,5

Comments

(1, 1, 2, 3, 5, 8, ...) convolved with (0, 0, 1, 2, 4, 7, ...) = (0, 0, 1, 3, 8, ...). - Gary W. Adamson, Aug 14 2010

References

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

N. J. A. Sloane, Table of n, a(n) for n=1..200

J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.

J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)

N. J. A. Sloane, Maple programs for counting rooted trees by height (after Riordan)

Index entries for sequences related to rooted trees

Index entries for sequences related to trees

Formula

a(n) = A001383(n) - A000041(n-1). - Christian G. Bower

Maple

# For Maple program see link.
with(combstruct):
ZL:= proc(m) local i; [T0, {seq(T||i=Prod(Z, Set(T||(i+1))), i=0..m-1), T||m=Z}, unlabeled] end: A000235:= n-> count(ZL(3), size=n)-count(ZL(2), size=n): seq(A000235(n), n=1..36); # Zerinvary Lajos, Sep 23 2007

Mathematica

m = 36; Rest @ CoefficientList[ Series[x*Product[(1-x^k)^(-PartitionsP[k-1]), {k, 1, m}], {x, 0, m}], x] - PartitionsP[Range[0, m-1]] (* Jean-François Alcover, Jul 05 2011, after Christian G. Bower *)

Crossrefs

Column h=3 of A034781.

Sequence in context: A328539 A078409 A036642 * A006478 A104187 A131051

Adjacent sequences: A000232 A000233 A000234 * A000236 A000237 A000238

Keyword

nonn

Author

N. J. A. Sloane

Status

approved