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, 18123840, 22151360, 26898080, 32463200, 38955840, 46495680

Offset: 0

Author

Zerinvary Lajos, Apr 27 2005

Keywords

• easy
• nonn

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

Examples

			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.
		

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).

Crossrefs

Cf. A000292, A062145.

Programs

Magma

A105939:= func< n | 20*Binomial(n+6,6) >;
[A105939(n): n in [0..40]]; // G. C. Greubel, Mar 11 2025

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 *)
20*Binomial[Range[0,40] +6,6] (* G. C. Greubel, Mar 11 2025 *)

SageMath

def A105939(n): return 20*binomial(n+6,6)
print([A105939(n) for n in range(41)]) # G. C. Greubel, Mar 11 2025

Formula

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

E.g.f.: (d/dx)^6 (x^3/3!)^2 * exp(x). - 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)

E.g.f.: (1/36)*(720 + 4320*x + 5400*x^2 + 2400*x^3 + 450*x^4 + 36*x^5 + x^6)*exp(x). - G. C. Greubel, Mar 11 2025

Extensions

More terms from Geoffrey Critzer, Sep 03 2013