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A105251 a(n) = binomial(n+4,n)*binomial(n+8,n).

%I A105251 #28 Mar 04 2025 08:12:25
%S A105251 1,45,675,5775,34650,162162,630630,2123550,6370650,17381650,43801758,
%T A105251 103169430,229265400,484306200,978496200,1900457064,3563356995,
%U A105251 6473226375,11428041625,19658764125,33026723730,54295634250,87501228750,138447123750,215362192500
%N A105251 a(n) = binomial(n+4,n)*binomial(n+8,n).
%H A105251 T. D. Noe, <a href="/A105251/b105251.txt">Table of n, a(n) for n = 0..1000</a>
%H A105251 <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 A105251 G.f.: (1+32*x+168*x^2+224*x^3+70*x^4)/(1-x)^13. - _Colin Barker_, Jan 21 2013
%F A105251 From _Amiram Eldar_, Sep 01 2022: (Start)
%F A105251 Sum_{n>=0} 1/a(n) = 640*Pi^2 - 13925728/2205.
%F A105251 Sum_{n>=0} (-1)^n/a(n) = 640*Pi^2/3 - 90112*log(2)/105 + 471984/1225. (End)
%e A105251 a(0): C(0+4,0)*C(0+8,0) = C(4,0)*C(8,0) = 1*1 = 1;
%e A105251 a(10): C(10+4,10)*C(10+8,10) = C(14,10)*(18,10) = 1001*43758 = 43801758.
%t A105251 f[n_] := Binomial[n + 4, n]Binomial[n + 8, n]; Table[ f[n], {n, 0, 22}] (* _Robert G. Wilson v_, Apr 20 2005 *)
%t A105251 LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1},{1,45,675,5775,34650,162162,630630,2123550,6370650,17381650,43801758,103169430,229265400},30] (* _Harvey P. Dale_, May 19 2024 *)
%o A105251 (Magma) [Binomial(n+4,n)*Binomial(n+8,n): n in [0..30]]; // _Vincenzo Librandi_, Jul 31 2015
%o A105251 (SageMath)
%o A105251 def A105251(n): return binomial(n+4,n)*binomial(n+8,n)
%o A105251 print([A105251(n) for n in range(31)]) # _G. C. Greubel_, Mar 04 2025
%Y A105251 Cf. A062264.
%K A105251 easy,nonn
%O A105251 0,2
%A A105251 _Zerinvary Lajos_, Apr 14 2005
%E A105251 More terms from _Robert G. Wilson v_, Apr 20 2005