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A107419 a(n) = binomial(n+4,4)*binomial(n+7,7).

%I A107419 #22 Mar 10 2025 05:07:55
%S A107419 1,40,540,4200,23100,99792,360360,1132560,3185325,8179600,19467448,
%T A107419 43439760,91706160,184497600,355816800,661028544,1187785665,
%U A107419 2071432440,3516320500,5824819000,9436206780,14978106000,23333661000,35728290000,53840548125,79942445856
%N A107419 a(n) = binomial(n+4,4)*binomial(n+7,7).
%H A107419 T. D. Noe, <a href="/A107419/b107419.txt">Table of n, a(n) for n = 0..1000</a>
%H A107419 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).
%F A107419 From _Chai Wah Wu_, Apr 10 2021: (Start)
%F A107419 a(n) = 12*a(n-1) - 66*a(n-2) + 220*a(n-3) - 495*a(n-4) + 792*a(n-5) - 924*a(n-6) + 792*a(n-7) - 495*a(n-8) + 220*a(n-9) - 66*a(n-10) + 12*a(n-11) - a(n-12) for n > 11.
%F A107419 G.f.: (1 + 28*x + 126*x^2 + 140*x^3 + 35*x^4)/(1 - x)^12. (End)
%F A107419 From _Amiram Eldar_, Sep 06 2022: (Start)
%F A107419 Sum_{n>=0} 1/a(n) = 392*Pi^2 - 870268/225.
%F A107419 Sum_{n>=0} (-1)^n/a(n) = 56*Pi^2/3 - 3584*log(2)/15 - 441/25. (End)
%e A107419 a(0) = C(0+4,4)*C(0+7,7) = C(4,4)*C(7,7) = 1*1 = 1.
%e A107419 a(6) = C(6+4,4)*C(6+7,7) = C(10,4)*C(13,7) = 210*1716 = 360360.
%t A107419 Table[Binomial[n+4,4]Binomial[n+7,7],{n,0,30}] (* _Harvey P. Dale_, Jul 27 2019 *)
%o A107419 (PARI) for(n=0,29,print1(binomial(n+4,4)*binomial(n+7,7),","))
%o A107419 (Magma)
%o A107419 A107419:= func< n | Binomial(n+4,n)*Binomial(n+7,n) >;
%o A107419 [A107419(n): n in [0..40]]; // _G. C. Greubel_, Mar 10 2025
%o A107419 (SageMath)
%o A107419 def A107419(n): return binomial(n+4,n)*binomial(n+7,n)
%o A107419 print([A107419(n) for n in range(41)]) # _G. C. Greubel_, Mar 10 2025
%Y A107419 Cf. A062145.
%K A107419 easy,nonn
%O A107419 0,2
%A A107419 _Zerinvary Lajos_, May 26 2005
%E A107419 Corrected and extended by _Rick L. Shepherd_, May 27 2005