A000065 -1 + number of partitions of n.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Number of graphs with n vertices of treewidth n-2. Reason: The complement of a graph with n vertices and treewidth >= n-2 cannot have P3 or K3 as a subgraph (Chlebı́ková 2002, Theorem 10), so every component of it is a star. - Martín Muñoz, Dec 31 2023

Examples

			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 + ...
		

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

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (first 199 terms from N. J. A. Sloane)
• Janka Chlebı́ková, The Structure of Obstructions to Treewidth and Pathwidth. Disc. Appl. Math., Vol. 120, no. 1-3 (2002), 61-71.
• Harmandeep Kaur and Muhammad Asif Rana, Partitions with unique largest part and their generating functions, arXiv:2506.11447 [math.CO], 2025.
• Vladimir Modrak and David 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.
• J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)

Crossrefs

A000041 - 1. A column of A058716. A diagonal of A263294.

Column h=2 of A034781.

Programs

Magma

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

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 *)

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)};

Formula

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

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

G.f.: Sum_{k>=2} x^k / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Sep 07 2021