A034827 a(n) = 2*binomial(n,4). (Formerly N0778)

0, 0, 0, 0, 2, 10, 30, 70, 140, 252, 420, 660, 990, 1430, 2002, 2730, 3640, 4760, 6120, 7752, 9690, 11970, 14630, 17710, 21252, 25300, 29900, 35100, 40950, 47502, 54810, 62930, 71920, 81840, 92752, 104720, 117810, 132090, 147630, 164502, 182780

Offset

0,5

Comments

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.

2,10,30,70, ... gives orchard crossing number of complete graph K_n. - Ralf Stephan, Mar 28 2003

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

Middle column of table on p. 6 of Feder and Garber. - Jonathan Vos Post, Apr 23 2009

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

From a(2), convolution of the oblong numbers (A002378) with the nonnegative numbers (A001477). - Bruno Berselli, Oct 24 2016

Also the number of 3-cycles in the n-triangular honeycomb bishop graph. - Eric W. Weisstein, Aug 10 2017

References

Charles Jordan, Calculus of Finite Differences, Chelsea, 1965, p. 449.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Links

Bruno Berselli, Table of n, a(n) for n = 0..1000

M. Aganagic, A. Klemm and C. Vafa, Disk Instantons, Mirror Symmetry and the Duality Web, arXiv:hep-th/0105045, 2001.

Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.

Elie Feder and David Garber, The Orchard crossing number of an abstract graph, arXiv:math/0303317 [math.CO], 2003-2009.

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), pp. 1917-1926.

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

Eric Weisstein's World of Mathematics, Graph Cycle.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = A096338(2*n-6) = 2*A000332(n), n>2. - R. J. Mathar, Nov 08 2010

G.f.: 2*x^4/(1-x)^5. - Colin Barker, Feb 29 2012

a(n) = Sum_{k=1..n-3} ( Sum_{i=1..k} i*(2*k-n+4) ). - Wesley Ivan Hurt, Sep 26 2013

E.g.f.: x^4*exp(x)/12. - G. C. Greubel, Feb 23 2017

From Amiram Eldar, Jul 19 2022: (Start)

Sum_{n>=4} 1/a(n) = 2/3.

Sum_{n>=4} (-1)^n/a(n) = 16*log(2) - 32/3. (End)

Maple

[seq(binomial(n, 4)*2, n=0..40)]; # Zerinvary Lajos, Jul 18 2006

Mathematica

CoefficientList[Series[2 x^4/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 20 2013 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 0, 0, 2}, 50] (* Harvey P. Dale, Jun 09 2016 *)
Table[2 Binomial[n, 4], {n, 0, 40}] (* Bruno Berselli, Oct 24 2016 *)
2 Binomial[Range[0, 20], 4] (* Eric W. Weisstein, Aug 10 2017 *)

Prog

(Magma) [2*Binomial(n, 4): n in [0..40]]; // Vincenzo Librandi, Oct 20 2013
(PARI) a(n)=2*binomial(n, 4) \\ Charles R Greathouse IV, Jun 23 2015

Crossrefs

A diagonal of A088617.

Cf. A033487, A050534, A060008.

Partial sums of A007290.

Cf. A001477, A002378.

Cf. A051843 (4-cycles in the triangular honeycomb bishop graph), A290775 (5-cycles), A290779 (6-cycles).

Sequence in context: A034262 A180499 A167214 * A328532 A051667 A106355

Adjacent sequences: A034824 A034825 A034826 * A034828 A034829 A034830

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved