cp's OEIS Frontend

A107417 a(n) = binomial(n+2,2)*binomial(n+5,5).

%I A107417 #19 Mar 10 2025 05:07:51
%S A107417 1,18,126,560,1890,5292,12936,28512,57915,110110,198198,340704,563108,
%T A107417 899640,1395360,2108544,3113397,4503114,6393310,8925840,12273030,
%U A107417 16642340,22281480,29484000,38595375,50019606,64226358,81758656,103241160,129389040,161017472,199051776
%N A107417 a(n) = binomial(n+2,2)*binomial(n+5,5).
%H A107417 G. C. Greubel, <a href="/A107417/b107417.txt">Table of n, a(n) for n = 0..1000</a>
%H A107417 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F A107417 From _Harvey P. Dale_, Feb 18 2012: (Start)
%F A107417 a(0)=1, a(1)=18, a(2)=126, a(3)=560, a(4)=1890, a(5)=5292, a(6)=12936, a(7)=28512, a(n)=8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)- 28*a(n-6)+ 8*a(n-7)-a(n-8).
%F A107417 G.f.: (1 + 10*x + 10*x^2)/(1-x)^8. (End)
%F A107417 From _Amiram Eldar_, Sep 08 2022: (Start)
%F A107417 Sum_{n>=0} 1/a(n) = 25*Pi^2/3 - 5845/72.
%F A107417 Sum_{n>=0} (-1)^n/a(n) = 205/8 - 5*Pi^2/2. (End)
%F A107417 E.g.f.: (1/240)*(240 + 4080*x + 10920*x^2 + 9400*x^3 + 3350*x^4 + 542*x^5 + 39*x^6 + x^7)*exp(x). - _G. C. Greubel_, Mar 10 2025
%e A107417 If n=0 then C(0+2,2)*C(0+5,5) = C(2,2)*C(5,5) = 1*1 = 1.
%e A107417 If n=3 then C(3+2,2)*C(3+5,5) = C(5,2)*C(8,5) = 10*56 = 560.
%t A107417 Table[Binomial[n+2,2]Binomial[n+5,5],{n,0,40}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{1,18,126,560,1890,5292,12936,28512},40] (* _Harvey P. Dale_, Feb 18 2012 *)
%o A107417 (PARI) for(n=0,40,print1(binomial(n+2,2)*binomial(n+5,5),","))
%o A107417 (Magma)
%o A107417 A107417:= func< n | Binomial(n+2,n)*Binomial(n+5,n) >;
%o A107417 [A107417(n): n in [0..40]]; // _G. C. Greubel_, Mar 10 2025
%o A107417 (SageMath)
%o A107417 def A107417(n): return binomial(n+2,n)*binomial(n+5,n)
%o A107417 print([A107417(n) for n in range(41)]) # _G. C. Greubel_, Mar 10 2025
%Y A107417 Cf. A062145.
%K A107417 easy,nonn
%O A107417 0,2
%A A107417 _Zerinvary Lajos_, May 26 2005
%E A107417 More terms from _Rick L. Shepherd_, May 27 2005