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A000094 Number of trees of diameter 4.
(Formerly M1350 N0518)
11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

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

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.

Index entries for sequences related to trees

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

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-Fran├žois Alcover, Feb 04 2016 *)

CROSSREFS

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

Sequence in context: A274523 A165189 A011842 * A182377 A058578 A261526

Adjacent sequences:  A000091 A000092 A000093 * A000095 A000096 A000097

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified August 16 08:52 EDT 2017. Contains 290623 sequences.