Search: id:A000712
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%I A000712 M1376 N0536
%S A000712 1,2,5,10,20,36,65,110,185,300,481,752,1165,1770,2665,3956,5822,8470,
%T A000712 12230,17490,24842,35002,49010,68150,94235,129512,177087,240840,326015,
%U A000712 439190,589128,786814,1046705,1386930,1831065,2408658,3157789,4126070
%N A000712 Number of partitions of n into parts of 2 kinds.
%C A000712 For n >= 1 a(n) is also the number of conjugacy classes in the automorphism 
               group of the n-dimensional hypercube. This automorphism group is 
               the wreath product of the cyclic group C_2 and the symmetric group 
               S_n, its order is in sequence A000165. - Dan Fux (dan.fux(AT)OpenGaia.com 
               or danfux(AT)OpenGaia.com), Nov 04 2001
%C A000712 Euler transform of period 1 sequence [2,2,2,...].
%C A000712 A006330(n)+A001523(n)=a(n). - Michael Somos, Jul 22 2003
%C A000712 Also, number of noncongruent matrices in GL_n(Z): each Jordan block can 
               only have +1 or -1 on the diagonal. - Michele Dondi (blazar(AT)lcm.mi.infn.it), 
               Jun 15 2004
%C A000712 a(n) = Sum (k(1)+1)*(k(2)+1)*...*(k(n)+1), where the sum is taken over 
               all (k(1),k(2),...,k(n)) such that k(1)+2*k(2)+...+n*k(n) = n, k(i)>
               =0, i=1..n, cf. A104510, A077285. - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Apr 21 2005
%C A000712 Convolution of partition numbers (A000041) with itself. - Graeme McRae 
               (g_m(AT)mcraefamily.com), Jun 07 2006
%C A000712 Number of one-to-one partial endofunctions on n unlabeled points. Connected 
               components are either cycles or "lines", hence two for each size. 
               - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Dec 28 2006
%C A000712 Equals A000716: (1, 3, 9, 22, 561, 108,...) convolved with A010815. A000716 
               = the number of partitions of n into parts of 3 kinds = the Euler 
               transform of [3,3,3,...]. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Oct 26 2008]
%D A000712 W. Y. C. Chen, K. Q. Ji and H. S. Wilf, BG-ranks and 2-cores, arXiv:math.CO/
               0605474.
%D A000712 H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, 
               Vol. 4, Cambridge Univ. Press, 1958, p. 90.
%D A000712 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
%D A000712 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000712 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000712 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see 
               Proposition 2.5.2 on page 78.
%H A000712 T. D. Noe, <a href="b000712.txt">Table of n, a(n) for n=0..500</a>
%H A000712 E. R. Canfield, C. D. Savage and H. S. Wilf, <a href="http://arXiv.org/
               abs/math.CO/0308061">Regularly spaced subsums of integer partitions</
               a>
%H A000712 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=129">
               Encyclopedia of Combinatorial Structures 129</a>
%H A000712 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%H A000712 <a href="Sindx_Pro.html#1mxtok">Index entries for expansions of Product_{k 
               >= 1} (1-x^k)^m</a>
%F A000712 G.f.: Product 1/(1-x^m)^2; m=1..inf. a(n) = sum k=0, ..., n p(k)*p(n-k).
%F A000712 a(n) ~ 1/12*3^(1/4)*n^(-5/4)*exp(2/3*3^(1/2)*pi*n^(1/2)) - Joe Keane 
               (jgk(AT)jgk.org), Sep 13 2002
%F A000712 G.f. : product(i=1, oo, (1+x^i)^(2*A001511(i))) (see A000041) - Jon Perry 
               (perry(AT)globalnet.co.uk), Jun 06 2004
%p A000712 with(combinat): A000712:=n->sum(numbpart(k)*numbpart(n-k),k=0..n): seq(A000712(n),
               n=0..37); (E. Deutsch)
%t A000712 CoefficientList[ Series[ Product[1/(1 - x^n)^2, {n, 40}], {x, 0, 37}], 
               x] (from Robert G. Wilson v Feb 03 2005)
%o A000712 (PARI) a(n)=local(X); if(n<0,0,X=x+x*O(x^n); polcoeff(1/eta(X)^2,n))
%Y A000712 Cf. A000165, A000041, A002107 (reciprocal of g.f.).
%Y A000712 Cf. A002720.
%Y A000712 A000716, A010815 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 26 
               2008]
%Y A000712 Sequence in context: A103928 A103929 A121597 this_sequence A032442 A102688 
               A001629
%Y A000712 Adjacent sequences: A000709 A000710 A000711 this_sequence A000713 A000714 
               A000715
%K A000712 nonn,easy,nice
%O A000712 0,2
%A A000712 N. J. A. Sloane (njas(AT)research.att.com).
%E A000712 More terms from Joe Keane (jgk(AT)jgk.org), Nov 17 2001
%E A000712 More terms from Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 15 2004

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