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A105249 a(n) = binomial(n+2,n)*binomial(n+6,n).

%I A105249 #31 Apr 07 2025 10:00:28
%S A105249 1,21,168,840,3150,9702,25872,61776,135135,275275,528528,965328,
%T A105249 1689324,2848860,4651200,7379904,11415789,17261937,25573240,37191000,
%U A105249 53183130,74890530,103980240,142506000,192976875,258434631,342540576,449672608,585033240,754769400
%N A105249 a(n) = binomial(n+2,n)*binomial(n+6,n).
%H A105249 G. C. Greubel, <a href="/A105249/b105249.txt">Table of n, a(n) for n = 0..1000</a>
%H A105249 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A105249 a(0)=1, a(1)=21, a(2)=168, a(3)=840, a(4)=3150, a(5)=9702, a(6)=25872, a(7)=61776, a(8)=135135, a(n) = 9*a(n-1) -36*a(n-2) +84*a(n-3) -126*a(n-4) +126*a(n-5) -84*a(n-6) +36*a(n-7) -9*a(n-8) +a(n-9). - _Harvey P. Dale_, Oct 08 2012
%F A105249 G.f.: (1+12*x+15*x^2)/(1-x)^9. - _Colin Barker_, Jan 21 2013
%F A105249 From _Amiram Eldar_, Sep 04 2022: (Start)
%F A105249 Sum_{n>=0} 1/a(n) = 12*Pi^2 - 5869/50.
%F A105249 Sum_{n>=0} (-1)^n/a(n) = 256*log(2)/5 - 4*Pi^2 + 371/75. (End)
%F A105249 E.g.f.: (1/1440)*(1440 + 28800*x + 91440*x^2 + 95520*x^3 + 42900*x^4 + 9312*x^5 + 1010*x^6 + 52*x^7 + x^8)*exp(x). - _G. C. Greubel_, Mar 04 2025
%e A105249 a(0): C(0+2,0)*C(0+6,0) = C(2,0)*C(6,0) = 1*1 = 1;
%e A105249 a(10): C(10+2,10)*C(10+6,10) = C(12,10)*C(16,10) = 66*8008 = 528528.
%t A105249 f[n_] := Binomial[n + 2, n]Binomial[n + 6, n]; Table[ f[n], {n, 0, 27}] (* _Robert G. Wilson v_, Apr 20 2005 *)
%t A105249 LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,21,168,840,3150,9702,25872,61776,135135},30] (* _Harvey P. Dale_, Oct 08 2012 *)
%o A105249 (Magma) [Binomial(n+2,n)*Binomial(n+6,n): n in [0..30]]; // _Vincenzo Librandi_, Jul 31 2015
%o A105249 (SageMath)
%o A105249 def A105249(n): return binomial(n+2,n)*binomial(n+6,n)
%o A105249 print([a105249(n) for n in range(31)]) # _G. C. Greubel_, Mar 04 2025
%Y A105249 Cf. A062264.
%K A105249 easy,nonn
%O A105249 0,2
%A A105249 _Zerinvary Lajos_, Apr 14 2005
%E A105249 More terms from _Robert G. Wilson v_, Apr 20 2005