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 A058716 Triangle T(n,k) giving number of nonisomorphic loopless matroids of rank k on n labeled points (n >= 0, 0 <= k <= n). 6
 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 6, 9, 4, 1, 0, 1, 8, 19, 16, 5, 1, 0, 1, 10, 33, 44, 25, 6, 1, 0, 1, 12, 51, 96, 85, 36, 7, 1, 0, 1, 14, 73, 180, 225, 146, 49, 8, 1, 0, 1, 16, 99, 304, 501, 456, 231, 64, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS A signed version is given by A119328. - Paul Barry, May 14 2006 LINKS 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. FORMULA From Paul Barry, May 14 2006: (Start) T(n,k) = Sum{i = 0..n} (-1)^(i-k) * C(n,i) * sum{j = 0..i-k} C(k,2j)*C(i-k,2j). Column k has g.f. (x/(1-x))^k * Sum{j = 0..k} C(k,2j) * x^(2j). (End) EXAMPLE Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:   1;   0, 1;   0, 1,  1;   0, 1,  2,  1;   0, 1,  4,  3,  1;   0, 1,  6,  9,  4,  1;   0, 1,  8, 19, 16,  5, 1;   0, 1, 10, 33, 44, 25, 6, 1;   ... MATHEMATICA t[n_, k_] := Sum[(-1)^(i-k)*Binomial[n, i]*Sum[Binomial[k, 2*j]*Binomial[i-k, 2*j], {j, 0, i-k}] , {i, 0, n}]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 21 2013 *) CROSSREFS Cf. A058717 (same except for border), A058710, A058711. Row sums give A058718. Diagonals give A000065, A058719. Sequence in context: A055277 A301422 A055340 * A119328 A048723 A088455 Adjacent sequences:  A058713 A058714 A058715 * A058717 A058718 A058719 KEYWORD nonn,tabl,nice AUTHOR N. J. A. Sloane, Dec 31 2000 EXTENSIONS Corrected and extended by Jean-François Alcover, Oct 21 2013 STATUS approved

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Last modified February 20 15:52 EST 2020. Contains 332078 sequences. (Running on oeis4.)