Search: id:A002865
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%I A002865 M0309 N0113
%S A002865 1,0,1,1,2,2,4,4,7,8,12,14,21,24,34,41,55,66,88,105,137,165,210,253,
%T A002865 320,383,478,574,708,847,1039,1238,1507,1794,2167,2573,3094,3660,4378,
%U A002865 5170,6153,7245,8591,10087,11914,13959,16424,19196,22519,26252,30701
%N A002865 Number of partitions of n that do not contain 1 as a part.
%C A002865 Also the number of partitions of n-1, n>=2, such that the least part 
               occurs exactly once. See A096373, A097091, A097092, A097093. - Robert 
               G. Wilson v Jul 24 2004 [Corrected by Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Feb 18 2009]
%C A002865 a(n) = A116449(n) + A116450(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Feb 16 2006
%C A002865 Number of partitions of n+1 where the number of parts is itself a part. 
               Take a partition of n (with k parts) which does not contain 1, remove 
               1 from each part and add a new part of size k+1. - Frank Adams-Watters 
               (FrankTAW(AT)Netscape.net), May 01 2006
%C A002865 Equals row sums of triangle A147768 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Nov 11 2008]
%C A002865 Contribution from Lewis Mammel (l_mammel(AT)att.net), Oct 06 2009: (Start)
%C A002865 a(n) is the number of sets of n disjoint pairs of 2n things,
%C A002865 called a pairing, disjoint with a given pairing ( A053871, )
%C A002865 that are unique under permutations preserving the given pairing.
%C A002865 Can be seen immediately from a graphical representation which must
%C A002865 decompose into even numbered cycles of 4 or more things, as connected
%C A002865 by pairs alternating between the pairings. Each thing is in a single 
               cycle,
%C A002865 so this is a partition of 2n into even parts greater than 2,
%C A002865 equivalent to a partition of n into parts greater than 1. (End)
%D A002865 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002865 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002865 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, 1964 (and various 
               reprintings), p. 836.
%D A002865 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, p*(n).
%D A002865 H. Gropp, On tactical configurations, regular bipartite graphs and (v,
               k,even)-designs, Discr. Math., 155 (1996), 81-98.
%D A002865 P. G. Tait, Scientific Papers, Cambridge Univ. Press, Vol. 1, 1898, Vol. 
               2, 1900, see Vol. 1, p. 334.
%H A002865 T. D. Noe, <a href="b002865.txt">Table of n, a(n) for n=0..1000</a>
%H A002865 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A002865 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=100">
               Encyclopedia of Combinatorial Structures 100</a>
%H A002865 J. L. Nicolas and A. Sarkozy, <a href="http://almira.math.u-bordeaux.fr/
               jtnb/2000-1/jtnb12-1.html#jourelec">On partitions without small parts</
               a>
%H A002865 <a href="Sindx_Par.html#partN">Index entries for related partition-counting 
               sequences</a>
%F A002865 G.f.: Product_{m>1} 1/(1-x^m).
%F A002865 a(0)=1, a(n)= p(n)-p(n-1), n>=1, with the partition numbers p(n) := A000041(n).
%F A002865 a(n) = A085811(n+2). - James Sellers, Dec 06 2005.
%F A002865 a(n) = Sum(A008284(n-k+1,k-1): 1<k<=floor((n+2)/2) for n>0. - Reinhard 
               Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 04 2007
%e A002865 a(6) = 4 from 6 = 4+2 = 3+3 = 2+2+2.
%p A002865 spec := [ B, {B=Set(Set(Z,card>1))}, unlabeled ]; [seq(combstruct[count](spec, 
               size=n), n=1..50)];
%p A002865 with(combstruct):ZL1:=[S,{S=Set(Cycle(Z,card>1))}, unlabeled]:seq(count(ZL1,
               size=n),n=0..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Sep 24 2007
%p A002865 G:={P=Set(Set(Atom,card>1))}:combstruct[gfsolve](G,unlabeled,x):seq(combstruct[count]([P,
               G,unlabeled],size=i),i=0..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Dec 16 2007
%p A002865 with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, unlabeled]; 
               end: A:=a(2):seq(count(A, size=n), n=0..50); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 11 2008
%t A002865 Table[ PartitionsP[n + 1] - PartitionsP[n], {n, -1, 50}] (from Robert 
               G. Wilson v Jul 24 2004)
%o A002865 (PARI) a(n)=if(n<0,0,polcoeff((1-x)/eta(x+x*O(x^n)),n))
%Y A002865 First differences of partition numbers A000041. Cf. A053445, A072380, 
               A081094, A081095.
%Y A002865 Pairwise sums seem to be in A027336.
%Y A002865 Essentially the same as A085811.
%Y A002865 Cf. A025147.
%Y A002865 A147768 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 11 2008]
%Y A002865 Sequence in context: A035979 A035989 A036000 this_sequence A085811 A014810 
               A026929
%Y A002865 Adjacent sequences: A002862 A002863 A002864 this_sequence A002866 A002867 
               A002868
%K A002865 nonn,easy,nice
%O A002865 0,5
%A A002865 N. J. A. Sloane (njas(AT)research.att.com).

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