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A105942 a(n) = binomial(n+6,6)*binomial(n+9,6).

%I A105942 #15 Mar 11 2025 04:37:55
%S A105942 84,1470,12936,77616,360360,1387386,4624620,13741728,37165128,
%T A105942 92912820,217273056,479693760,1007356896,2024399916,3912705720,
%U A105942 7303717344,13213962300,23241027810,39841761960,66720654000,109363854600,175763337750,277386503940,430459323840
%N A105942 a(n) = binomial(n+6,6)*binomial(n+9,6).
%H A105942 G. C. Greubel, <a href="/A105942/b105942.txt">Table of n, a(n) for n = 0..1000</a>
%H A105942 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
%F A105942 G.f.: 42*(1 + x)*(2 + 7*x + 2*x^2)/(1-x)^13. - _Harvey P. Dale_, Sep 14 2012
%F A105942 a(0)=84, a(1)=1470, a(2)=12936, a(3)=77616, a(4)=360360, a(5)=1387386, a(6)=4624620, a(7)=13741728, a(8)=37165128, a(9)=92912820, a(10)=217273056, a(11)=479693760, a(12)=1007356896, a(n) = 13*a(n-1) -78*a(n-2) +286*a(n-3) -715*a(n-4) +1287*a(n-5) -1716*a(n-6) +1716*a(n-7) -1287*a(n-8) +715*a(n-9) -286*a(n-10) +78*a(n-11) -13*a(n-12) +a(n-13). - _Harvey P. Dale_, Sep 14 2012
%F A105942 From _Amiram Eldar_, Sep 08 2022: (Start)
%F A105942 Sum_{n>=0} 1/a(n) = 10446039/3920 - 270*Pi^2.
%F A105942 Sum_{n>=0} (-1)^n/a(n) = 82911/560 - 15*Pi^2. (End)
%e A105942 If n=0 then C(0+6,0)*C(0+9,6) = C(6,0)*C(9,6) = 1*84 = 84.
%e A105942 If n=6 then C(6+6,6)*C(6+9,6) = C(12,6)*C(15,6) = 924*5005 = 4624620.
%t A105942 Table[Binomial[n+6,n]Binomial[n+9,6],{n,0,30}] (* or *) CoefficientList[ Series[-((42 (x+1) (x (2 x+7)+2))/(x-1)^13),{x,0,30}],x] (* _Harvey P. Dale_, Sep 14 2012 *)
%o A105942 (Magma)
%o A105942 A105942:= func< n | Binomial(n+6,6)*Binomial(n+9,6)/42 >;
%o A105942 [A105942(n): n in [0..40]]; // _G. C. Greubel_, Mar 11 2025
%o A105942 (SageMath)
%o A105942 def A105942(n): return binomial(n+6,6)*binomial(n+9,6)
%o A105942 print([A105942(n) for n in range(41)]) # _G. C. Greubel_, Mar 11 2025
%Y A105942 Cf. A062145.
%K A105942 easy,nonn
%O A105942 0,1
%A A105942 _Zerinvary Lajos_, Apr 27 2005
%E A105942 Corrected and extended by _Harvey P. Dale_, Sep 14 2012