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%I M1012 N0379 #71 Jan 01 2024 09:11:43
%S 0,0,1,2,4,6,10,14,21,29,41,55,76,100,134,175,230,296,384,489,626,791,
%T 1001,1254,1574,1957,2435,3009,3717,4564,5603,6841,8348,10142,12309,
%U 14882,17976,21636,26014,31184,37337,44582,53173,63260,75174,89133,105557,124753
%N -1 + number of partitions of n.
%C a(n+1) is the number of noncongruent n-dimensional integer-sided simplices with diameter n. - _Sascha Kurz_, Jul 26 2004
%C Also, the number of partitions of n into parts each less than n.
%C Also, the number of distinct types of equation which can be derived from the equation [n,0,0] not including itself. (Ince)
%C Also, the number of rooted trees on n+1 nodes with height exactly 2.
%C 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
%C 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
%D E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944, p. 498; MR0010757.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A000065/b000065.txt">Table of n, a(n) for n = 0..1000</a> (first 199 terms from N. J. A. Sloane)
%H J. Chlebı́ková, <a href="https://doi.org/10.1016/S0166-218X(01)00281-5">The Structure of Obstructions to Treewidth and Pathwidth</a>. Disc. Appl. Math., Vol. 120, no. 1-3 (2002), 61-71.
%H V. Modrak and D. Marton, <a href="http://dx.doi.org/10.3390/e15104285">Development of Metrics and a Complexity Scale for the Topology of Assembly Supply Chains</a>, Entropy 2013, 15, 4285-4299
%H J. Riordan, <a href="http://dx.doi.org/10.1147/rd.45.0473">Enumeration of trees by height and diameter</a>, IBM J. Res. Dev. 4 (1960), 473-478.
%H J. Riordan, <a href="/A007401/a007401_8.pdf">The enumeration of trees by height and diameter</a>, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
%F a(n) = A026820(n,n-1) for n>1. - _Reinhard Zumkeller_, Jan 21 2010
%F 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
%F G.f.: Sum_{k>=2} x^k / Product_{j=1..k} (1 - x^j). - _Ilya Gutkovskiy_, Sep 07 2021
%e 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 + ...
%p 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
%p ZL :=[S, {S = Set(Cycle(Z),1 < card)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=0..45); # _Zerinvary Lajos_, Mar 25 2008
%t nn=40;CoefficientList[Series[Product[1/(1-x^i),{i,1,nn}]-1/(1-x),{x,0,nn}],x] (* _Geoffrey Critzer_, Oct 28 2012 *)
%t PartitionsP[Range[0,50]]-1 (* _Harvey P. Dale_, Aug 24 2013 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( 1 / eta(x + x*O(x^n)), n) - 1)};
%o (PARI) {a(n) = if( n<0, 0, numbpart(n) - 1)};
%o (Magma) [NumberOfPartitions(n)-1: n in [0..50]]; // _Vincenzo Librandi_, Aug 25 2013
%Y A000041 - 1. A column of A058716. A diagonal of A263294.
%Y Column h=2 of A034781.
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_
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