%I #36 Mar 09 2022 11:40:02
%S 20,140,560,1680,4200,9240,18480,34320,60060,100100,160160,247520,
%T 371280,542640,775200,1085280,1492260,2018940,2691920,3542000,4604600,
%U 5920200,7534800,9500400,11875500,14725620
%N a(n) = binomial(n+3,3)*binomial(n+6,3).
%C 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
%H Michael De Vlieger, <a href="/A105939/b105939.txt">Table of n, a(n) for n = 0..10000</a>
%H Ömür Deveci and Anthony G. Shannon, <a href="https://doi.org/10.20948/mathmontis-2021-50-4">Some aspects of Neyman triangles and Delannoy arrays</a>, Mathematica Montisnigri (2021) Vol. L, 36-43.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F G.f.: 20/(1-x)^7. - _Colin Barker_, Jun 06 2012
%F With offset = 6, e.g.f.: exp(x)*x^3/3!*x^3/3!. - _Geoffrey Critzer_, Sep 03 2013
%F a(n) = A000292(n+1)*A000292(n+4) = 20*A000579(n+6). - _R. J. Mathar_, Nov 30 2015
%F From _Amiram Eldar_, Jan 06 2021: (Start)
%F Sum_{n>=0} 1/a(n) = 3/50.
%F Sum_{n>=0} (-1)^n/a(n) = 48*log(2)/5 - 661/100. (End)
%e If n=0 then C(0+3,0)*C(0+6,3) = C(3,0)*C(6,3) = 1*20 = 20.
%e If n=8 then C(8+3,8)*C(8+6,3) = C(11,8)*C(14,3) = 165*364 = 60060.
%t 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 *)
%t 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 *)
%Y Cf. A000292, A062145.
%K easy,nonn
%O 0,1
%A _Zerinvary Lajos_, Apr 27 2005
%E More terms from _Geoffrey Critzer_, Sep 03 2013