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A105947 a(n) = C(n+4,4) * C(n+6,6).

%I A105947 #23 Feb 22 2025 03:31:03
%S A105947 1,35,420,2940,14700,58212,194040,566280,1486485,3578575,8016008,
%T A105947 16893240,33786480,64574160,118605600,210327264,361499985,604167795,
%U A105947 984569740,1568220500,2446423980,3744526500,5632263000,8336601000,12157543125,17487410031,24834191760
%N A105947 a(n) = C(n+4,4) * C(n+6,6).
%H A105947 G. C. Greubel, <a href="/A105947/b105947.txt">Table of n, a(n) for n = 0..1000</a>
%H A105947 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
%F A105947 G.f.: (1 + 24*x + 90*x^2 + 80*x^3 + 15*x^4)/(1-x)^11. - _Colin Barker_, Jan 28 2013
%F A105947 From _Wesley Ivan Hurt_, Jan 27 2022: (Start)
%F A105947 a(n) = (17280 + 78336*n + 152376*n^2 + 167780*n^3 + 116150*n^4 + 52983*n^5 +
%F A105947    16173*n^6 + 3270*n^7 + 420*n^8 + 31*n^9 + n^10)/17280.
%F A105947 a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11). (End)
%F A105947 From _Amiram Eldar_, Sep 08 2022: (Start)
%F A105947 Sum_{n>=0} 1/a(n) = 224*Pi^2 - 55244/25.
%F A105947 Sum_{n>=0} (-1)^n/a(n) = 12*Pi^2 + 512*log(2)/5 - 4711/25. (End)
%e A105947 If n=0 then C(0+6,0)*C(0+4,4) = C(6,0)*C(4,4) = 1*1 = 1.
%e A105947 If n=10 then C(10+6,10)*C(10+4,4) = C(16,10)*C(14,4) = 8008*1001 = 8016008.
%t A105947 Table[Binomial[n+6,n]Binomial[n+4,4],{n,0,30}] (* _Harvey P. Dale_, May 21 2014 *)
%o A105947 (Magma)
%o A105947 A105947:= func< n | Binomial(n+4,4)*Binomial(n+6,6) >;
%o A105947 [A105947(n): n in [0..40]]; // _G. C. Greubel_, Feb 22 2025
%o A105947 (SageMath)
%o A105947 def A105947(n): return binomial(n+4,4)*binomial(n+6,6)
%o A105947 print([A105947(n) for n in range(41)]) # _G. C. Greubel_, Feb 22 2025
%Y A105947 Cf. A062196.
%K A105947 easy,nonn
%O A105947 0,2
%A A105947 _Zerinvary Lajos_, Apr 27 2005
%E A105947 Terms from a(8) onwards corrected by _Colin Barker_, Jan 28 2013
%E A105947 Second example corrected by _Colin Barker_, Jan 28 2013