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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

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).

Programs

Magma

[2*Binomial(n,4): n in [0..40]]; // Vincenzo Librandi, Oct 20 2013

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 *)

PARI

a(n)=2*binomial(n,4) \\ Charles R Greathouse IV, Jun 23 2015

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)