cp's OEIS Frontend

A110207 a(n) = sum of cubes of numbers < 2^n having exactly floor(n/2) + 1 ones in their binary expansion.

%I A110207 #9 Oct 04 2024 07:41:34
%S A110207 1,27,368,6615,88536,1449198,20078192,320944275,4584724120,
%T A110207 72867715074,1064153845776,16896536592390,250629464211504,
%U A110207 3980364331323996,59709362473177824,948742244639103915,14352114907032903000
%N A110207 a(n) = sum of cubes of numbers < 2^n having exactly floor(n/2) + 1 ones in their binary expansion.
%C A110207 a(n) equals the largest term in row n of triangle A110205.
%H A110207 G. C. Greubel, <a href="/A110207/b110207.txt">Table of n, a(n) for n = 1..825</a>
%F A110207 a(n) = (8^n-1)/7*C(n-3, floor(n/2)) + ((2^n-1)*(4^n-1)-(8^n-1)/7)*C(n-3, floor(n/2)-1) + (2^n-1)^3*C(n-3, floor(n/2)-2).
%t A110207 b[n_, k_]:= Binomial[n-3, Floor[n/2]-k];
%t A110207 f[n_]:= (8^n-1)/7*(b[n,0] - b[n,1]) + (2^n-1)^2*((2^n+1)*b[n,1] + (2^n - 1)*b[n,2]);
%t A110207 A110207[n_]:= If[n<3, f[n]/2, f[n]];
%t A110207 Table[A110207[n], {n,30}] (* _G. C. Greubel_, Oct 03 2024 *)
%o A110207 (PARI) {a(n)=(8^n-1)/7*binomial(n-3,n\2)+((2^n-1)*(4^n-1)-(8^n-1)/7)*binomial(n-3,n\2-1) +(2^n-1)^3*binomial(n-3,n\2-2)}
%o A110207 (Magma)
%o A110207 b:= func< n,k | Binomial(n-3, Floor(n/2) - k) >;
%o A110207 A110207:= func< n | (8^n-1)/7*(b(n,0) - b(n,1)) + (2^n-1)^2*((2^n+1)*b(n,1) + (2^n-1)*b(n,2)) >;
%o A110207 [A110207(n): n in [1..30]]; // _G. C. Greubel_, Oct 03 2024
%o A110207 (SageMath)
%o A110207 def b(n,k): return binomial(n-3, (n//2) - k)
%o A110207 def A110207(n): return (8^n-1)/7*(b(n,0) - b(n,1)) + (2^n-1)^2*((2^n+1)*b(n,1) + (2^n-1)*b(n,2))
%o A110207 [A110207(n) for n in range(1,31)] # _G. C. Greubel_, Oct 03 2024
%Y A110207 Cf. A110205 (triangle), A110206 (row sums).
%K A110207 nonn
%O A110207 1,2
%A A110207 _Paul D. Hanna_, Jul 16 2005