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A104474 a(n) = binomial(n+3,3)*binomial(n+7,3).

%I A104474 #29 May 25 2025 16:32:16
%S A104474 35,224,840,2400,5775,12320,24024,43680,75075,123200,194480,297024,
%T A104474 440895,638400,904400,1256640,1716099,2307360,3059000,4004000,5180175,
%U A104474 6630624,8404200,10556000,13147875,16248960,19936224,24295040,29419775
%N A104474 a(n) = binomial(n+3,3)*binomial(n+7,3).
%H A104474 G. C. Greubel, <a href="/A104474/b104474.txt">Table of n, a(n) for n = 0..1000</a>
%H A104474 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A104474 a(n) = (1/36)*(n+1)*(n+2)*(n+3)*(n+5)*(n+6)*(n+7).
%F A104474 G.f.: (35 - 21*x + 7*x^2 - x^3)/(1-x)^7. - _R. J. Mathar_, Nov 30 2015
%F A104474 a(n) = A000292(n+1)*A000292(n+5). - _R. J. Mathar_, Nov 30 2015
%F A104474 From _Amiram Eldar_, Jan 06 2021: (Start)
%F A104474 Sum_{n>=0} 1/a(n) = 7/200.
%F A104474 Sum_{n>=0} (-1)^n/a(n) = 1/40. (End)
%F A104474 From _G. C. Greubel_, Mar 05 2025: (Start)
%F A104474 a(n) = 140*A000580(n+7)/(n+4).
%F A104474 E.g.f.: (1/36)*(1260 + 6804*x + 7686*x^2 + 3102*x^3 + 531*x^4 + 39*x^5 + x^6)*exp(x). (End)
%e A104474 a(0): C(0+3,3)*C(0+7,3) = C(3,3)*C(7,3) = 1*35 = 35.
%e A104474 a(7): C(7+3,3)*C(7+7,3) = C(10,3)*(14,3) = 120*364 = 43680.
%t A104474 f[n_] := Binomial[n + 3, 3]Binomial[n + 7, 3]; Table[ f[n], {n, 0, 28}] (* _Robert G. Wilson v_, Apr 20 2005 *)
%t A104474 LinearRecurrence[{7,-21,35,-35,21,-7,1},{35,224,840,2400,5775,12320,24024},40] (* _Harvey P. Dale_, May 25 2025 *)
%o A104474 (PARI) vector(30, n, n--; binomial(n+3,3)*binomial(n+7,3)) \\ _Michel Marcus_, Jul 31 2015
%o A104474 (Magma) [Binomial(n+3,3)*Binomial(n+7,3): n in [0..30]]; // _Vincenzo Librandi_, Jul 31 2015
%o A104474 (SageMath)
%o A104474 def A104474(n): return 140*binomial(n+7,7)//(n+4)
%o A104474 print([A104474(n) for n in range(31)]) # _G. C. Greubel_, Mar 05 2025
%Y A104474 Cf. A000292, A000580, A062264.
%K A104474 easy,nonn
%O A104474 0,1
%A A104474 _Zerinvary Lajos_, Apr 18 2005
%E A104474 More terms from _Robert G. Wilson v_, Apr 20 2005