

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

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 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



FORMULA

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


EXAMPLE

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
combinat[numbpart](n1)n+1 ;


MATHEMATICA

t=Table[PartitionsP[n]n, {n, 0, 45}];
ReplacePart[t, 0, 1]


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



