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%I #22 Sep 04 2022 04:06:21
%S 1,21,168,840,3150,9702,25872,61776,135135,275275,528528,965328,
%T 1689324,2848860,4651200,7379904,11415789,17261937,25573240,37191000,
%U 53183130,74890530,103980240,142506000,192976875,258434631,342540576,449672608,585033240,754769400
%N a(n) = binomial(n+2,n)*binomial(n+6,n).
%H <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 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 G.f.: -(15*x^2+12*x+1)/(x-1)^9. - _Colin Barker_, Jan 21 2013
%F From _Amiram Eldar_, Sep 04 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 12*Pi^2 - 5869/50.
%F Sum_{n>=0} (-1)^n/a(n) = 256*log(2)/5 - 4*Pi^2 + 371/75. (End)
%e a(0): C(0+2,0)*C(0+6,0) = C(2,0)*C(6,0) = 1*1 = 1;
%e a(10): C(10+2,10)*C(10+6,10) = C(12,10)*C(16,10) = 66*8008 = 528528.
%t f[n_] := Binomial[n + 2, n]Binomial[n + 6, n]; Table[ f[n], {n, 0, 27}] (* _Robert G. Wilson v_, Apr 20 2005 *)
%t 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 (Magma) [Binomial(n+2,n)*Binomial(n+6,n): n in [0..30]]; // _Vincenzo Librandi_, Jul 31 2015
%Y Cf. A062264.
%K easy,nonn
%O 0,2
%A _Zerinvary Lajos_, Apr 14 2005
%E More terms from _Robert G. Wilson v_, Apr 20 2005
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