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A007695 Cardinalities of Sperner families on 1,...,n.

%I A007695 M2466 #28 Nov 29 2023 11:41:41
%S A007695 2,3,5,10,26,96,553,5461,100709,3718354,289725509,49513793526,
%T A007695 19089032278261,16951604697397302,35231087224279091310,
%U A007695 173550485517380958360611,2047581288200721764035942914
%N A007695 Cardinalities of Sperner families on 1,...,n.
%C A007695 Also number of f-vectors for simplicial complexes on at most n vertices.
%D A007695 S. Johnson, Upper bounds for constant weight error correcting codes, Discrete Math., 3 (1972), 109-124.
%D A007695 D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.3 (p. 743).
%D A007695 D. E. Knuth, Art of Computer Programming, Vol. 4, Section 7.3, to appear.
%D A007695 S. Linusson, The number of M-sequences and f-vectors, Combinatorica, 19 (1999), 255-266.
%D A007695 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007695 D. Knuth, <a href="/A007695/a007695_2.pdf">Email to N. J. A. Sloane, Aug. 1994</a>.
%H A007695 Tamon Stephen and Timothy Yusun, <a href="http://arxiv.org/abs/1209.4623">Counting inequivalent monotone Boolean functions</a>, arXiv preprint arXiv:1209.4623 [cs.DS], 2012.
%t A007695 c[ 0, 0 ]=1; c[ 0, 1 ]=1; kap[ 0, 0 ]=0; f[ n_ ] := Block[ {s=2, r, d, k, j}, For[ r=1, r<=n, r++, d=s; k=r; j=0; s=0;
%t A007695 For[ x=0, x<=Binomial[ n, r ], x++, If[ x>=Binomial[ k, r ], k++, 0 ]; kap[ r, x ]=If[ x==0, 0, Binomial[ k-1, r-1 ]+kap[ r-1, x-Binomial[ k-1, r ] ] ];
%t A007695 While[ j<kap[ r, x ], d -= c[ r-1, j ]; j++ ]; c[ r, x ]=d; s += d; ] ]; s ]
%Y A007695 This is the limiting form of A011828-A011833.
%Y A007695 Cf. A001405.
%K A007695 nonn,nice
%O A007695 0,1
%A A007695 _N. J. A. Sloane_, _Don Knuth_
%E A007695 Entry revised by _N. J. A. Sloane_, Sep 03 2011