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A105252 a(n) = binomial(n+5,n)*binomial(n+9,n).

%I A105252 #33 Mar 04 2025 08:12:30
%S A105252 1,60,1155,12320,90090,504504,2312310,9060480,31286970,97337240,
%T A105252 277411134,733649280,1818838840,4261894560,9502285320,20271542016,
%U A105252 41572498275,82281899700,157706974425,293570877600,532097215650,941124327000,1627522854750,2756636064000
%N A105252 a(n) = binomial(n+5,n)*binomial(n+9,n).
%H A105252 T. D. Noe, <a href="/A105252/b105252.txt">Table of n, a(n) for n = 0..1000</a>
%H A105252 <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).
%F A105252 G.f.: (1+45*x+360*x^2+840*x^3+630*x^4+126*x^5)/(1-x)^15. - _Colin Barker_, Jan 21 2013
%F A105252 From _Amiram Eldar_, Sep 01 2022: (Start)
%F A105252 Sum_{n>=0} 1/a(n) = 7425*Pi^2/2 - 114902691/3136.
%F A105252 Sum_{n>=0} (-1)^n/a(n) = 28960047/15680 - 255*Pi^2/4 - 12288*log(2)/7. (End)
%e A105252 a(0): C(0+5,0)*C(0+9,0) = C(5,0)*C(9,0) = 1*1 = 1;
%e A105252 a(10): C(10+5,10)*C(10+9,10) = C(15,10)*(19,10) = 3003*92378 = 277411134.
%t A105252 f[n_] := Binomial[n + 5, n]Binomial[n + 9, n]; Table[ f[n], {n, 0, 20}] (* _Robert G. Wilson v_, Apr 20 2005 *)
%o A105252 (Magma) [Binomial(n+5,n)*Binomial(n+9,n): n in [0..30]]; // _Vincenzo Librandi_, Jul 31 2015
%o A105252 (Python)
%o A105252 A105252_list, m = [], [2002, -4433, 3487, -1133, 127, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
%o A105252 for _ in range(10**2):
%o A105252     A105252_list.append(m[-1])
%o A105252     for i in range(14):
%o A105252         m[i+1] += m[i] # _Chai Wah Wu_, Dec 15 2015
%o A105252 (SageMath)
%o A105252 def A105252(n): return binomial(n+5,n)*binomial(n+9,n)
%o A105252 print([A105252(n) for n in range(31)]) # _G. C. Greubel_, Mar 04 2025
%Y A105252 Cf. A062264.
%K A105252 easy,nonn
%O A105252 0,2
%A A105252 _Zerinvary Lajos_, Apr 14 2005
%E A105252 More terms from _Robert G. Wilson v_, Apr 20 2005