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A084625 Binomial transform of A084624.

%I A084625 #6 Mar 24 2023 17:18:43
%S A084625 1,3,8,21,55,143,366,919,2265,5491,13125,31000,72485,168042,386709,
%T A084625 884161,2009742,4543830,10222264,22891099,51041560,113359224,
%U A084625 250839510,553173006,1216070081,2665518207,5826533103,12703217438,27628250142
%N A084625 Binomial transform of A084624.
%H A084625 G. C. Greubel, <a href="/A084625/b084625.txt">Table of n, a(n) for n = 0..1000</a>
%F A084625 a(n) = Sum_{k=0..n} C(n, k)*floor(C(k+5, 5)/C(k+2, 2)).
%F A084625 a(n) = 2^n + Sum_{k=0..n} binomial(n,k)*floor(k*(k^2 +12*k +47)/60). - _G. C. Greubel_, Mar 24 2023
%t A084625 a[n_]:= a[n]= 2^n +Sum[Binomial[n,j]*Floor[j*(j^2+12*j+47)/60], {j,0, n}];
%t A084625 Table[a[n], {n,0,50}] (* _G. C. Greubel_, Mar 24 2023 *)
%o A084625 (Magma)
%o A084625 A084625:= func< n | (&+[Binomial(n,j)*Floor(Binomial(j+5,3)/10): j in [0..n]]) >;
%o A084625 [A084625(n): n in [0..50]]; // _G. C. Greubel_, Mar 24 2023
%o A084625 (SageMath)
%o A084625 def A084625(n): return sum(binomial(n,j)*(binomial(j+5,3)//10) for j in range(n+1))
%o A084625 [A084625(n) for n in range(51)] # _G. C. Greubel_, Mar 24 2023
%Y A084625 Cf. A084624, A084626, A084627, A084628, A084630, A084631.
%K A084625 easy,nonn
%O A084625 0,2
%A A084625 _Paul Barry_, Jun 01 2003