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
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
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)
J. Chlebı́ková, The Structure of Obstructions to Treewidth and Pathwidth. Disc. Appl. Math., Vol. 120, no. 1-3 (2002), 61-71.
V. Modrak and 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.
J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
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
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved