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

%I A105939 #44 Mar 11 2025 17:06:00
%S A105939 20,140,560,1680,4200,9240,18480,34320,60060,100100,160160,247520,
%T A105939 371280,542640,775200,1085280,1492260,2018940,2691920,3542000,4604600,
%U A105939 5920200,7534800,9500400,11875500,14725620,18123840,22151360,26898080,32463200,38955840,46495680
%N A105939 a(n) = binomial(n+3,3)*binomial(n+6,3).
%C A105939 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 A105939 Michael De Vlieger, <a href="/A105939/b105939.txt">Table of n, a(n) for n = 0..10000</a>
%H A105939 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 A105939 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A105939 G.f.: 20/(1-x)^7. - _Colin Barker_, Jun 06 2012
%F A105939 E.g.f.: (d/dx)^6 (x^3/3!)^2 * exp(x). - _Geoffrey Critzer_, Sep 03 2013
%F A105939 a(n) = A000292(n+1)*A000292(n+4) = 20*A000579(n+6). - _R. J. Mathar_, Nov 30 2015
%F A105939 From _Amiram Eldar_, Jan 06 2021: (Start)
%F A105939 Sum_{n>=0} 1/a(n) = 3/50.
%F A105939 Sum_{n>=0} (-1)^n/a(n) = 48*log(2)/5 - 661/100. (End)
%F A105939 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
%e A105939 If n=0 then C(0+3,0)*C(0+6,3) = C(3,0)*C(6,3) = 1*20 = 20.
%e A105939 If n=8 then C(8+3,8)*C(8+6,3) = C(11,8)*C(14,3) = 165*364 = 60060.
%t A105939 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 A105939 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 *)
%t A105939 20*Binomial[Range[0,40] +6,6] (* _G. C. Greubel_, Mar 11 2025 *)
%o A105939 (Magma)
%o A105939 A105939:= func< n | 20*Binomial(n+6,6) >;
%o A105939 [A105939(n): n in [0..40]]; // _G. C. Greubel_, Mar 11 2025
%o A105939 (SageMath)
%o A105939 def A105939(n): return 20*binomial(n+6,6)
%o A105939 print([A105939(n) for n in range(41)]) # _G. C. Greubel_, Mar 11 2025
%Y A105939 Cf. A000292, A062145.
%K A105939 easy,nonn
%O A105939 0,1
%A A105939 _Zerinvary Lajos_, Apr 27 2005
%E A105939 More terms from _Geoffrey Critzer_, Sep 03 2013