A000094 Number of trees of diameter 4.

0, 0, 0, 0, 1, 2, 5, 8, 14, 21, 32, 45, 65, 88, 121, 161, 215, 280, 367, 471, 607, 771, 980, 1232, 1551, 1933, 2410, 2983, 3690, 4536, 5574, 6811, 8317, 10110, 12276, 14848, 17941, 21600, 25977, 31146, 37298, 44542, 53132, 63218, 75131, 89089

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Number of partitions of n-1 with at least two parts of size 2 or larger. - Franklin T. Adams-Watters, Jan 13 2006

Also equal to the number of partitions p of n-1 such that max(p)-min(p) > 1. Example: a(7)=5 because we have [5,1],[4,2],[4,1,1],[3,2,1] and [3,1,1,1]. - Giovanni Resta, Feb 06 2006

Also number of partitions of n-1 with at least two parts that are smaller than the largest part. Example: a(7)=5 because we have [4,1,1],[3,2,1],[3,1,1,1],[2,2,1,1,1] and [2,1,1,1,1]. - Emeric Deutsch, May 01 2006

Also number of regions of n-1 that do not contain 1 as a part, n >= 2 (cf. A186114, A206437). - Omar E. Pol, Dec 01 2011

Also rank of the last region of n-1 multiplied by -1, n >= 2 (cf. A194447). - Omar E. Pol, Feb 11 2012

Also sum of ranks of the regions of n-1 that contain emergent parts, n >= 2 (cf. A182699). For the definition of "regions of n" see A206437. - Omar E. Pol, Feb 21 2012

Examples

			From _Gus Wiseman_, Apr 12 2019: (Start)
The a(5) = 1 through a(9) = 14 partitions of n-1 with at least two parts of size 2 or larger, or non-hooks, are the following. The Heinz numbers of these partitions are given by A105441.
  (22)  (32)   (33)    (43)     (44)
        (221)  (42)    (52)     (53)
               (222)   (322)    (62)
               (321)   (331)    (332)
               (2211)  (421)    (422)
                       (2221)   (431)
                       (3211)   (521)
                       (22111)  (2222)
                                (3221)
                                (3311)
                                (4211)
                                (22211)
                                (32111)
                                (221111)
The a(5) = 1 through a(9) = 14 partitions of n-1 whose maximum part minus minimum part is at least 2 are the following. The Heinz numbers of these partitions are given by A307516.
  (31)  (41)   (42)    (52)     (53)
        (311)  (51)    (61)     (62)
               (321)   (331)    (71)
               (411)   (421)    (422)
               (3111)  (511)    (431)
                       (3211)   (521)
                       (4111)   (611)
                       (31111)  (3221)
                                (3311)
                                (4211)
                                (5111)
                                (32111)
                                (41111)
                                (311111)
The a(5) = 1 through a(9) = 14 partitions of n-1 with at least two parts that are smaller than the largest part are the following. The Heinz numbers of these partitions are given by A307517.
  (211)  (311)   (321)    (322)     (422)
         (2111)  (411)    (421)     (431)
                 (2211)   (511)     (521)
                 (3111)   (3211)    (611)
                 (21111)  (4111)    (3221)
                          (22111)   (3311)
                          (31111)   (4211)
                          (211111)  (5111)
                                    (22211)
                                    (32111)
                                    (41111)
                                    (221111)
                                    (311111)
                                    (2111111)
(End)
		

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

• Christian G. Bower, Table of n, a(n) for n = 1..500
• 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)
• Miloslav Znojil, Perturbation theory near degenerate exceptional points, arXiv:2008.00479 [math-ph], 2020.
• Index entries for sequences related to trees

Crossrefs

Cf. A000041, A206437, A034853, A000147 (diameter 5).

Cf. A006918, A083751, A084835, A105441, A115720, A257990, A307516, A307517, A325164.

Programs

Maple

g:=x/product(1-x^j,j=1..70)-x-x^2/(1-x)^2: gser:=series(g,x=0,48): seq(coeff(gser,x,n),n=1..46); # Emeric Deutsch, May 01 2006
A000094 := proc(n)
    combinat[numbpart](n-1)-n+1 ;
end proc: # R. J. Mathar, May 17 2016

Mathematica

t=Table[PartitionsP[n]-n,{n,0,45}];
ReplacePart[t,0,1]
(* Clark Kimberling, Mar 05 2012 *)
CoefficientList[1/QPochhammer[x]-x/(1-x)^2-1+O[x]^50, x] (* Jean-François Alcover, Feb 04 2016 *)

Formula

a(n+1) = A000041(n)-n for n>0. - John W. Layman

G.f.: x/product(1-x^j,j=1..infinity)-x-x^2/(1-x)^2. - Emeric Deutsch, May 01 2006

G.f.: sum(sum(x^(i+j+1)/product(1-x^k, k=i..j), i=1..j-2), j=3..infinity). - Emeric Deutsch, May 01 2006

a(n+1) = Sum_{m=1..n} A083751(m). - Gregory Gerard Wojnar, Oct 13 2020

Extensions

More terms from Franklin T. Adams-Watters, Jan 13 2006