

A000094


Number of trees of diameter 4.
(Formerly M1350 N0518)


14



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
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OFFSET

1,6


COMMENTS

Number of partitions of n1 with at least two parts of size 2 or larger.  Franklin T. AdamsWatters, Jan 13 2006
Also equal to the number of partitions p of n1 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 n1 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 n1 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 n1 multiplied by 1, n >= 2 (cf. A194447).  Omar E. Pol, Feb 11 2012
Also sum of ranks of the regions of n1 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), 473478.
J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473478. (Annotated scanned copy)
Miloslav Znojil, Perturbation theory near degenerate exceptional points, arXiv:2008.00479 [mathph], 2020.
Index entries for sequences related to trees


FORMULA

a(n+1) = A000041(n)n for n>0.  John W. Layman
G.f.: x/product(1x^j,j=1..infinity)xx^2/(1x)^2.  Emeric Deutsch, May 01 2006
G.f.: sum(sum(x^(i+j+1)/product(1x^k, k=i..j), i=1..j2), j=3..infinity).  Emeric Deutsch, May 01 2006
a(n+1) = Sum_{m=1..n} A083751(m).  Gregory Gerard Wojnar, Oct 13 2020


EXAMPLE

From Gus Wiseman, Apr 12 2019: (Start)
The a(5) = 1 through a(9) = 14 partitions of n1 with at least two parts of size 2 or larger, or nonhooks, 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 n1 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 n1 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)


MAPLE

g:=x/product(1x^j, j=1..70)xx^2/(1x)^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](n1)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/(1x)^21+O[x]^50, x] (* JeanFrançois Alcover, Feb 04 2016 *)


CROSSREFS

Cf. A000041, A206437, A034853, A000147 (diameter 5).
Cf. A006918, A083751, A084835, A105441, A115720, A257990, A307516, A307517, A325164.
Sequence in context: A165189 A358055 A011842 * A182377 A327380 A330378
Adjacent sequences: A000091 A000092 A000093 * A000095 A000096 A000097


KEYWORD

nonn


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Franklin T. AdamsWatters, Jan 13 2006


STATUS

approved



