%I A000065 M1012 N0379
%S A000065 0,0,1,2,4,6,10,14,21,29,41,55,76,100,134,175,230,296,384,489,626,791,
%T A000065 1001,1254,1574,1957,2435,3009,3717,4564,5603,6841,8348,10142,12309,
%U A000065 14882,17976,21636,26014,31184,37337,44582,53173,63260,75174,89133
%N A000065 -1 + number of partitions of n.
%C A000065 a(n+1) is the number of noncongruent n-dimensional integer-sided simplices 
               with diameter n. - Sascha Kurz (sascha.kurz(AT)uni-byreuth.de), Jul 
               26 2004
%C A000065 Also, the number of partitions of n into parts each less than n.
%C A000065 Also, the number of distinct types of equation which can be derived from 
               the equation [n,0,0] not including itself. (Ince)
%C A000065 Also, the number of rooted trees on n nodes with height exactly 2.
%C A000065 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 (callan(AT)stat.wisc.edu), 
               Nov 29 2007
%D A000065 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000065 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000065 E. L. Ince, Ordinary Differential Equations, Dover Publications, New 
               York, 1944, p. 498; MR0010757.
%D A000065 J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. 
               Dev. 4 (1960), 473-478.
%H A000065 N. J. A. Sloane, <a href="b000065.txt">Table of n, a(n) for n=0..199</
               a>
%p A000065 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 (zerinvarylajos(AT)yahoo.com), 
               Dec 05 2007
%p A000065 ZL :=[S, {S = Set(Cycle(Z),1 < card)}, unlabelled]: seq(combstruct[count](ZL, 
               size=n), n=0..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 25 2008
%o A000065 (PARI) a(n)=if(n<0,0,polcoeff(1/eta(x+x*O(x^n)),n)-1)
%o A000065 (PARI) a(n)=if(n<0, 0, numbpart(n)-1)
%Y A000065 A000041 - 1. A diagonal of A058716.
%Y A000065 Sequence in context: A103259 A082380 A136460 this_sequence A023499 A103445 
               A001747
%Y A000065 Adjacent sequences: A000062 A000063 A000064 this_sequence A000066 A000067 
               A000068
%K A000065 nonn,easy
%O A000065 0,4
%A A000065 N. J. A. Sloane (njas(AT)research.att.com).