|
|
A000065
|
|
-1 + number of partitions of n.
(Formerly M1012 N0379)
|
|
45
|
|
|
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
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
|
|
|
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
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 *)
|
|
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)};
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|