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A104676 a(n) = binomial(n+2,2) * binomial(n+7,2).

%I A104676 #29 Mar 01 2025 11:16:54
%S A104676 21,84,216,450,825,1386,2184,3276,4725,6600,8976,11934,15561,19950,
%T A104676 25200,31416,38709,47196,57000,68250,81081,95634,112056,130500,151125,
%U A104676 174096,199584,227766,258825,292950,330336,371184,415701,464100,516600,573426,634809,700986
%N A104676 a(n) = binomial(n+2,2) * binomial(n+7,2).
%H A104676 G. C. Greubel, <a href="/A104676/b104676.txt">Table of n, a(n) for n = 0..1000</a>
%H A104676 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A104676 From _R. J. Mathar_, Nov 29 2015: (Start)
%F A104676 a(n) = A000217(n+1) * A000217(n+6).
%F A104676 G.f.: 3*(7 - 7*x + 2*x^2)/(1-x)^5. (End)
%F A104676 a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - _Wesley Ivan Hurt_, Jan 25 2022
%F A104676 From _Amiram Eldar_, Aug 30 2022: (Start)
%F A104676 Sum_{n>=0} 1/a(n) = 7/100.
%F A104676 Sum_{n>=0} (-1)^n/a(n) = 7/180. (End)
%F A104676 E.g.f.: (1/4)*(84 + 252*x + 138*x^2 + 22*x^3 + x^4)*exp(x). - _G. C. Greubel_, Mar 01 2025
%e A104676 If n=0 then C(0+2,0+0)*C(0+7,2) = C(2,0)*C(7,2) = 1*21 = 21.
%e A104676 If n=8 then C(8+2,8+0)*C(8+7,2) = C(10,8)*C(15,2) = 45*105 = 4725.
%p A104676 A104676:=n->binomial(n+2,2)*binomial(n+7,2): seq(A104676(n), n=0..50); # _Wesley Ivan Hurt_, Mar 30 2017
%t A104676 Table[Binomial[n + 2, 2] Binomial[n + 7, 2], {n, 0, 37}] (* _Michael De Vlieger_, Nov 29 2015 *)
%o A104676 (PARI) a(n) = binomial(n+2,2)*binomial(n+7,2); \\ _Michel Marcus_, Nov 29 2015
%o A104676 (Magma)
%o A104676 A104676:= func< n | Binomial(n+2,2)*Binomial(n+7,2) >;
%o A104676 [A104676(n): n in [0..50]]; // _G. C. Greubel_, Mar 01 2025
%o A104676 (SageMath)
%o A104676 def A104676(n): return binomial(n+2,2)*binomial(n+7,2)
%o A104676 print([A104676(n) for n in range(51)]) # _G. C. Greubel_, Mar 01 2025
%Y A104676 Cf. A000217, A062190.
%Y A104676 Subsequence of A085780.
%K A104676 easy,nonn
%O A104676 0,1
%A A104676 _Zerinvary Lajos_, Apr 22 2005