A263294 -id:A263294 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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 + ...

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).

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)

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

Column h=2 of A034781.

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

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

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 *)

{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)};
```

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

A362907 Number of graphs on n unlabeled nodes with treewidth 3.

Original entry on oeis.org

0, 0, 0, 0, 1, 6, 53, 501, 5889, 81786

Offset: 0

Martin Rubey, May 09 2023

			There is a single graph of treewidth 3 on 4 vertices, which is the complete graph.

Wikipedia, Treewidth.

Column k=3 of A263294.

[sum(1 for g in graphs(n) if g.treewidth() == 3) for n in range(9)]

A362908 Number of graphs on n unlabeled nodes with treewidth 2.

Original entry on oeis.org

0, 0, 0, 1, 4, 17, 72, 323, 1639, 9203

Offset: 0

Martin Rubey, May 09 2023

			There is a single graph of treewidth 2 on 3 vertices, which is the complete graph.

Wikipedia, Treewidth.

Column 2 of A263294.

[sum(1 for g in graphs(n) if g.treewidth() == 2) for n in range(9)]

A387046 Triangle read by rows: T(n,k) is the number of unlabeled simple graphs with n vertices and treedepth k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 5, 1, 1, 6, 16, 10, 1, 1, 10, 47, 75, 22, 1, 1, 14, 129, 466, 386, 47, 1, 1, 21, 332, 2751, 6512, 2615, 113, 1, 1, 29, 816, 14298, 96913, 138336, 23982, 292, 1, 1, 41, 1951, 68951, 1159664, 5804406, 4652868, 316417, 868, 1, 1, 55, 4557, 318789, 12070626, 170635411, 580118945, 249848040, 5998477, 2962, 1

Offset: 1

Kolja Kühn, Aug 14 2025

The treedepth of a graph is the minimum height of a rooted forest whose closure contains the graph.
It is also the vertex ranking number.
A graph without edges has treedepth 1, any other graph where each connected component is a star or an isolated vertex has treedepth 2.
The complete graph on n vertices has treedepth n.
Values are computed by combining the programs nauty by Brendan McKay and Adolfo Piperno and Bute by James Trimble.

			Triangle begins:
  1;
  1, 1;
  1, 2, 1;
  1, 4, 5, 1;
  1, 6, 16, 10, 1;
  1, 10, 47, 75, 22, 1;
  1, 14, 129, 466, 386, 47, 1;
  1, 21, 332, 2751, 6512, 2615, 113, 1;
  1, 29, 816, 14298, 96913, 138336, 23982, 292, 1;
  1, 41, 1951, 68951, 1159664, 5804406, 4652868, 316417, 868, 1;
  1, 55, 4557, 318789, 12070626, 170635411, 580118945, 249848040, 5998477, 2962, 1;
  ...

J. Nešetřil and P. Ossona de Mendez, Sparsity: Graphs, Structures, and Algorithms, Springer, 2012.

Brendan McKay and Adolfo Piperno, nauty
James Trimble, Bute
Wikipedia, Tree-depth

Row sums are A000088.

Cf. A263294.

A387431 Triangle read by rows: T(n,k) is the number of unlabeled simple connected graphs with n vertices and treedepth k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 10, 9, 1, 0, 1, 25, 64, 21, 1, 0, 1, 62, 380, 363, 46, 1, 0, 1, 137, 2196, 6103, 2567, 112, 1, 0, 1, 294, 10963, 89989, 135673, 23868, 291, 1, 0, 1, 599, 51051, 1055752, 5663404, 4628772, 316124, 867, 1, 0, 1, 1187, 230003, 10805643, 164689853, 575441978, 249531330, 5997608, 2961, 1

Offset: 1

Kolja Kühn, Aug 29 2025

The treedepth of a graph is the minimum height of a rooted forest whose closure contains the graph.
It is also the vertex ranking number.
A graph without edges has treedepth 1, any other graph where each connected component is a star or an isolated vertex has treedepth 2.
The complete graph on n vertices has treedepth n.
Values are computed by combining the programs nauty by Brendan McKay and Adolfo Piperno and Bute by James Trimble.

			Triangle begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 4, 1;
  0, 1, 10, 9, 1;
  0, 1, 25, 64, 21, 1;
  0, 1, 62, 380, 363, 46, 1;
  0, 1, 137, 2196, 6103, 2567, 112, 1;
  0, 1, 294, 10963, 89989, 135673, 23868, 291, 1;
  0, 1, 599, 51051, 1055752, 5663404, 4628772, 316124, 867, 1;
  0, 1, 1187, 230003, 10805643, 164689853, 575441978, 249531330, 5997608, 2961, 1;
  ...

J. Nešetřil and P. Ossona de Mendez, Sparsity: Graphs, Structures, and Algorithms, Springer, 2012.

Brendan McKay and Adolfo Piperno, nauty
James Trimble, Bute
Wikipedia, Tree-depth

Row sums are A001349.

Cf. A387046 (analogous sequence including disconnected graphs), A263294.

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A000065 -1 + number of partitions of n.

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A362907 Number of graphs on n unlabeled nodes with treewidth 3.

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A362908 Number of graphs on n unlabeled nodes with treewidth 2.

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A387046 Triangle read by rows: T(n,k) is the number of unlabeled simple graphs with n vertices and treedepth k.

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A387431 Triangle read by rows: T(n,k) is the number of unlabeled simple connected graphs with n vertices and treedepth k.

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