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%I #47 Sep 05 2023 15:12:19
%S 0,0,1,3,7,13,23,37,58,87,128,183,259,359,493,668,898,1194,1578,2067,
%T 2693,3484,4485,5739,7313,9270,11705,14714,18431,22995,28598,35439,
%U 43787,53929,66238,81120,99096,120732,146746,177930,215267,259849,313022,376282
%N a(n) = p(0) + p(1) + ... + p(n) - n - 1, where p = partition numbers, A000041.
%C Number of non-isomorphic rank-2 matroids over S_n.
%C Starting (1, 3, 7, 13, ...) = row sums of triangle A171239. - _Gary W. Adamson_, Dec 05 2009
%D Acketa, Dragan M. "On the enumeration of matroids of rank-2." Zbornik radova Prirodnomatematickog fakulteta-Univerzitet u Novom Sadu 8 (1978): 83-90. - _N. J. A. Sloane_, Dec 04 2022
%H Alois P. Heinz, <a href="/A058682/b058682.txt">Table of n, a(n) for n = 0..10000</a> (first 501 terms from Muniru A Asiru)
%H W. M. B. Dukes, <a href="http://www.stp.dias.ie/~dukes/matroid.html">Tables of matroids</a>.
%H W. M. B. Dukes, <a href="https://web.archive.org/web/20030208144026/http://www.stp.dias.ie/~dukes/phd.html">Counting and Probability in Matroid Theory</a>, Ph.D. Thesis, Trinity College, Dublin, 2000.
%H W. M. B. Dukes, <a href="https://arxiv.org/abs/math/0411557">The number of matroids on a finite set</a>, arXiv:math/0411557 [math.CO], 2004.
%H W. M. B. Dukes, <a href="https://www.emis.de/journals/SLC/wpapers/s51dukes.html">On the number of matroids on a finite set</a>, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g.
%H Markus Kirchweger, Manfred Scheucher, and Stefan Szeider, <a href="https://doi.org/10.4230/LIPIcs.SAT.2022.4">A SAT Attack on Rota's Basis Conjecture</a>, Leibniz International Proceedings in Informatics (LIPIcs 2022) Vol. 236, 4:1-4:18.
%H Dillon Mayhew and Gordon F. Royle, <a href="https://arxiv.org/abs/math/0702316">Matroids with nine elements</a>, arXiv:math/0702316 [math.CO], 2007 (see p. 7).
%H Dillon Mayhew and Gordon F. Royle, <a href="https://doi.org/10.1016/j.jctb.2007.07.005">Matroids with nine elements</a>, J. Combin. Theory Ser. B 98(2) (2008), 415-431.
%H <a href="/index/Mat#matroid">Index entries for sequences related to matroids</a>
%F G.f.: -1/(1 - x)^2 + (1/(1 - x))*Product_{k>=1} 1/(1 - x^k). - _Ilya Gutkovskiy_, Aug 10 2018
%p a:= proc(n) option remember; `if`(n<2, 0,
%p combinat[numbpart](n)+a(n-1)-1)
%p end:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Oct 10 2019
%t With[{s = PartitionsP /@ Range[0, 40]}, MapIndexed[Total@ Take[s, First@ #2] - First@ #2 &, s]] (* _Michael De Vlieger_, Sep 04 2017 *)
%t With[{nn=50},#[[2]]-#[[1]]&/@Thread[{Range[0,nn],Accumulate[PartitionsP[Range[0,nn]]]}]]-1 (* _Harvey P. Dale_, Sep 05 2023 *)
%o (GAP) List([1..41],n->Sum([1..n-1],k->NrPartitions(k)-1)); # _Muniru A Asiru_, Aug 10 2018
%Y Column k=2 of A053534.
%Y Cf. A000041, A000065 (first differences), A000070.
%Y Cf. A171239. - _Gary W. Adamson_, Dec 05 2009
%K nonn
%O 0,4
%A _N. J. A. Sloane_, Dec 30 2000
%E Name clarified by _Ilya Gutkovskiy_, Aug 10 2018
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