A058682 a(n) = p(0) + p(1) + ... + p(n) - n - 1, where p = partition numbers, A000041.
0, 0, 1, 3, 7, 13, 23, 37, 58, 87, 128, 183, 259, 359, 493, 668, 898, 1194, 1578, 2067, 2693, 3484, 4485, 5739, 7313, 9270, 11705, 14714, 18431, 22995, 28598, 35439, 43787, 53929, 66238, 81120, 99096, 120732, 146746, 177930, 215267, 259849, 313022, 376282
Offset: 0
Keywords
References
- 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
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 501 terms from Muniru A Asiru)
- W. M. B. Dukes, Tables of matroids.
- W. M. B. Dukes, Counting and Probability in Matroid Theory, Ph.D. Thesis, Trinity College, Dublin, 2000.
- W. M. B. Dukes, The number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
- W. M. B. Dukes, On the number of matroids on a finite set, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g.
- Markus Kirchweger, Manfred Scheucher, and Stefan Szeider, A SAT Attack on Rota's Basis Conjecture, Leibniz International Proceedings in Informatics (LIPIcs 2022) Vol. 236, 4:1-4:18.
- Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, arXiv:math/0702316 [math.CO], 2007 (see p. 7).
- Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, J. Combin. Theory Ser. B 98(2) (2008), 415-431.
- Index entries for sequences related to matroids
Crossrefs
Programs
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GAP
List([1..41],n->Sum([1..n-1],k->NrPartitions(k)-1)); # Muniru A Asiru, Aug 10 2018
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Maple
a:= proc(n) option remember; `if`(n<2, 0, combinat[numbpart](n)+a(n-1)-1) end: seq(a(n), n=0..50); # Alois P. Heinz, Oct 10 2019 -
Mathematica
With[{s = PartitionsP /@ Range[0, 40]}, MapIndexed[Total@ Take[s, First@ #2] - First@ #2 &, s]] (* Michael De Vlieger, Sep 04 2017 *) With[{nn=50},#[[2]]-#[[1]]&/@Thread[{Range[0,nn],Accumulate[PartitionsP[Range[0,nn]]]}]]-1 (* Harvey P. Dale, Sep 05 2023 *)
Formula
G.f.: -1/(1 - x)^2 + (1/(1 - x))*Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Aug 10 2018
Extensions
Name clarified by Ilya Gutkovskiy, Aug 10 2018
Comments