A105939 a(n) = binomial(n+3,3)*binomial(n+6,3).

20, 140, 560, 1680, 4200, 9240, 18480, 34320, 60060, 100100, 160160, 247520, 371280, 542640, 775200, 1085280, 1492260, 2018940, 2691920, 3542000, 4604600, 5920200, 7534800, 9500400, 11875500, 14725620

Offset

0,1

Comments

a(n) is the number of ordered pairs (A,B) of size 3 subsets of {1,2,...,n+6} such that A and B are disjoint. - Geoffrey Critzer, Sep 03 2013

Links

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

G.f.: 20/(1-x)^7. - Colin Barker, Jun 06 2012

With offset = 6, e.g.f.: exp(x)*x^3/3!*x^3/3!. - Geoffrey Critzer, Sep 03 2013

a(n) = A000292(n+1)*A000292(n+4) = 20*A000579(n+6). - R. J. Mathar, Nov 30 2015

From Amiram Eldar, Jan 06 2021: (Start)

Sum_{n>=0} 1/a(n) = 3/50.

Sum_{n>=0} (-1)^n/a(n) = 48*log(2)/5 - 661/100. (End)

Example

If n=0 then C(0+3,0)*C(0+6,3) = C(3,0)*C(6,3) = 1*20 = 20.
If n=8 then C(8+3,8)*C(8+6,3) = C(11,8)*C(14,3) = 165*364 = 60060.

Mathematica

nn=25; f[x_]:=Exp[x](x^3/3!)^2; Range[0, nn]!CoefficientList[Series[a=f''''''[x], {x, 0, nn}], x] (* Geoffrey Critzer, Sep 03 2013 *)
Table[Binomial[n+3, 3]Binomial[n+6, 3], {n, 0, 30}] (* or *) LinearRecurrence[ {7, -21, 35, -35, 21, -7, 1}, {20, 140, 560, 1680, 4200, 9240, 18480}, 30] (* Harvey P. Dale, Mar 09 2022 *)

Crossrefs

Cf. A000292, A062145.

Sequence in context: A236988 A358865 A134382 * A054389 A253003 A293932

Adjacent sequences: A105936 A105937 A105938 * A105940 A105941 A105942

Keyword

easy,nonn

Author

Zerinvary Lajos, Apr 27 2005

Extensions

More terms from Geoffrey Critzer, Sep 03 2013

Status

approved