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A000065 -1 + number of partitions of n.
(Formerly M1012 N0379)
27
0, 0, 1, 2, 4, 6, 10, 14, 21, 29, 41, 55, 76, 100, 134, 175, 230, 296, 384, 489, 626, 791, 1001, 1254, 1574, 1957, 2435, 3009, 3717, 4564, 5603, 6841, 8348, 10142, 12309, 14882, 17976, 21636, 26014, 31184, 37337, 44582, 53173, 63260, 75174, 89133, 105557, 124753 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

a(n+1) is the number of noncongruent n-dimensional integer-sided simplices with diameter n. - Sascha Kurz, Jul 26 2004

Also, the number of partitions of n into parts each less than n.

Also, the number of distinct types of equation which can be derived from the equation [n,0,0] not including itself. (Ince)

Also, the number of rooted trees on n+1 nodes with height exactly 2.

Also, the number of partitions (of any positive integer) whose sum + length is <= n. Example: a(5) = 6 counts 4, 3, 21, 2, 11, 1. Proof: Given a partition of n other than the all 1s partition, subtract 1 from each part and then drop the zeros. This is a bijection to the partitions with sum + length <= n. - David Callan, Nov 29 2007

a(n) = A026820(n,n-1) for n>1. - Reinhard Zumkeller, Jan 21 2010

REFERENCES

E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944, p. 498; MR0010757.

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 and Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (first 199 terms from N. J. A. Sloane)

V. Modrak, D. Marton, Development of Metrics and a Complexity Scale for the Topology of Assembly Supply Chains, Entropy 2013, 15, 4285-4299

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

FORMULA

G.f.: x*G(0)/(x-1) where G(k) =  1 - 1/(1-x^(k+1))/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 23 2013

EXAMPLE

G.f. = x^2 + 2*x^3 + 4*x^4 + 6*x^5 + 10*x^6 + 14*x^7 + 21*x^8 + 29*x^9 + ...

MAPLE

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:A:=n -> count(ZL(2), size=n)-count(ZL(1), size=n): seq(A(n), n=1..46); # Zerinvary Lajos, Dec 05 2007

ZL :=[S, {S = Set(Cycle(Z), 1 < card)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=0..45); # Zerinvary Lajos, Mar 25 2008

MATHEMATICA

nn=40; CoefficientList[Series[Product[1/(1-x^i), {i, 1, nn}]-1/(1-x), {x, 0, nn}], x]  (* Geoffrey Critzer, Oct 28 2012 *)

PartitionsP[Range[0, 50]]-1 (* Harvey P. Dale, Aug 24 2013 *)

PROG

(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / eta(x + x*O(x^n)), n) - 1)};

(PARI) {a(n) = if( n<0, 0, numbpart(n) - 1)};

(MAGMA) [NumberOfPartitions(n)-1: n in [0..50]]; // Vincenzo Librandi, Aug 25 2013

CROSSREFS

A000041 - 1. A diagonal of A058716.

Column h=2 of A034781.

Sequence in context: A082380 A238871 A136460 * A237758 A023499 A103445

Adjacent sequences:  A000062 A000063 A000064 * A000066 A000067 A000068

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 18 19:41 EDT 2017. Contains 290762 sequences.