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%I N0778 #103 Jul 19 2022 05:46:47
%S 0,0,0,0,2,10,30,70,140,252,420,660,990,1430,2002,2730,3640,4760,6120,
%T 7752,9690,11970,14630,17710,21252,25300,29900,35100,40950,47502,
%U 54810,62930,71920,81840,92752,104720,117810,132090,147630,164502,182780
%N a(n) = 2*binomial(n,4).
%C Also number of ways to insert two pairs of parentheses into a string of n-4 letters (allowing empty pairs of parentheses). E.g., there are 30 ways for 2 letters. Cf. A002415.
%C 2,10,30,70, ... gives orchard crossing number of complete graph K_n. - _Ralf Stephan_, Mar 28 2003
%C If Y is a 2-subset of an n-set X then, for n>=4, a(n-1) is the number of 4-subsets and 5-subsets of X having exactly one element in common with Y. - _Milan Janjic_, Dec 28 2007
%C Middle column of table on p. 6 of Feder and Garber. - _Jonathan Vos Post_, Apr 23 2009
%C Number of pairs of non-intersecting lines when each of n points around a circle is joined to every other point by straight lines. A pair of lines is considered non-intersecting if the lines do not intersect in either the interior or the boundary of a circle. - _Melvin Peralta_, Feb 05 2016
%C From a(2), convolution of the oblong numbers (A002378) with the nonnegative numbers (A001477). - _Bruno Berselli_, Oct 24 2016
%C Also the number of 3-cycles in the n-triangular honeycomb bishop graph. - _Eric W. Weisstein_, Aug 10 2017
%D Charles Jordan, Calculus of Finite Differences, Chelsea, 1965, p. 449.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%H Bruno Berselli, <a href="/A034827/b034827.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Aganagic, A. Klemm and C. Vafa, <a href="http://arXiv.org/abs/hep-th/0105045">Disk Instantons, Mirror Symmetry and the Duality Web</a>, arXiv:hep-th/0105045, 2001.
%H Steven Edwards and William Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Griffiths/griffiths51.html">On Generalized Delannoy Numbers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
%H Elie Feder and David Garber, <a href="http://arxiv.org/abs/math/0303317">The Orchard crossing number of an abstract graph</a>, arXiv:math/0303317 [math.CO], 2003-2009.
%H S. M. Losanitsch, <a href="http://dx.doi.org/10.1002/cber.189703002144">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber. 30 (1897), pp. 1917-1926.
%H S. M. Losanitsch, <a href="/A000602/a000602_1.pdf">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = A096338(2*n-6) = 2*A000332(n), n>2. - _R. J. Mathar_, Nov 08 2010
%F G.f.: 2*x^4/(1-x)^5. - _Colin Barker_, Feb 29 2012
%F a(n) = Sum_{k=1..n-3} ( Sum_{i=1..k} i*(2*k-n+4) ). - _Wesley Ivan Hurt_, Sep 26 2013
%F E.g.f.: x^4*exp(x)/12. - _G. C. Greubel_, Feb 23 2017
%F From _Amiram Eldar_, Jul 19 2022: (Start)
%F Sum_{n>=4} 1/a(n) = 2/3.
%F Sum_{n>=4} (-1)^n/a(n) = 16*log(2) - 32/3. (End)
%p [seq(binomial(n,4)*2,n=0..40)]; # _Zerinvary Lajos_, Jul 18 2006
%t CoefficientList[Series[2 x^4/(1 - x)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 20 2013 *)
%t LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 0, 0, 2}, 50] (* _Harvey P. Dale_, Jun 09 2016 *)
%t Table[2 Binomial[n, 4], {n, 0, 40}] (* _Bruno Berselli_, Oct 24 2016 *)
%t 2 Binomial[Range[0, 20], 4] (* _Eric W. Weisstein_, Aug 10 2017 *)
%o (Magma) [2*Binomial(n,4): n in [0..40]]; // _Vincenzo Librandi_, Oct 20 2013
%o (PARI) a(n)=2*binomial(n,4) \\ _Charles R Greathouse IV_, Jun 23 2015
%Y A diagonal of A088617.
%Y Cf. A033487, A050534, A060008.
%Y Partial sums of A007290.
%Y Cf. A001477, A002378.
%Y Cf. A051843 (4-cycles in the triangular honeycomb bishop graph), A290775 (5-cycles), A290779 (6-cycles).
%K nonn,easy
%O 0,5
%A _N. J. A. Sloane_
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