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A107418 a(n) = binomial(n+3,3)*binomial(n+6,6).

%I A107418 #17 Mar 10 2025 05:07:47
%S A107418 1,28,280,1680,7350,25872,77616,205920,495495,1101100,2290288,4504864,
%T A107418 8446620,15193920,26356800,44279424,72299997,115079580,179012680,
%U A107418 272734000,407737330,599124240,866502000,1235052000,1736791875,2412056556,3311225568,4496726080,6045343480
%N A107418 a(n) = binomial(n+3,3)*binomial(n+6,6).
%H A107418 Robert Israel, <a href="/A107418/b107418.txt">Table of n, a(n) for n = 0..10000</a>
%H A107418 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F A107418 G.f.: (1 + 18*x + 45*x^2 + 20*x^3)/(1-x)^10. - _Robert Israel_, Feb 24 2017
%F A107418 From _Amiram Eldar_, Sep 06 2022: (Start)
%F A107418 Sum_{n>=0} 1/a(n) = 63*Pi^2 - 124149/200.
%F A107418 Sum_{n>=0} (-1)^n/a(n) = 3*Pi^2/2 + 1344*log(2)/5 - 40031/200. (End)
%e A107418 If n=0 then C(0+3,3)*C(0+6,6) = C(3,3)*C(6,6) = 1*1 = 1.
%e A107418 If n=8 then C(8+3,3)*C(8+6,6) = C(11,3)*C(14,6) = 165*3003 = 495495.
%p A107418 seq(binomial(n+3,3)*binomial(n+6,6),n=0..100); # _Robert Israel_, Feb 24 2017
%t A107418 a[n_] := Binomial[n + 3, 3] * Binomial[n + 6, 6]; Array[a, 30, 0] (* _Amiram Eldar_, Sep 06 2022 *)
%o A107418 (PARI) for(n=0,29,print1(binomial(n+3,3)*binomial(n+6,6),","))
%o A107418 (Magma)
%o A107418 A107418:= func< n | Binomial(n+3,n)*Binomial(n+6,n) >;
%o A107418 [A107418(n): n in [0..40]]; // _G. C. Greubel_, Mar 10 2025
%o A107418 (SageMath)
%o A107418 def A107418(n): return binomial(n+3,n)*binomial(n+6,n)
%o A107418 print([A107418(n) for n in range(41)]) # _G. C. Greubel_, Mar 10 2025
%Y A107418 Cf. A062145.
%K A107418 easy,nonn
%O A107418 0,2
%A A107418 _Zerinvary Lajos_, May 26 2005
%E A107418 Corrected and extended by _Rick L. Shepherd_, May 27 2005