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

1, 27, 368, 6615, 88536, 1449198, 20078192, 320944275, 4584724120, 72867715074, 1064153845776, 16896536592390, 250629464211504, 3980364331323996, 59709362473177824, 948742244639103915, 14352114907032903000

Offset: 1

Author

Paul D. Hanna, Jul 16 2005

Keywords

• nonn

Comments

a(n) equals the largest term in row n of triangle A110205.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..825

Crossrefs

Cf. A110205 (triangle), A110206 (row sums).

Programs

Magma

b:= func< n,k | Binomial(n-3, Floor(n/2) - k) >;
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)) >;
[A110207(n): n in [1..30]]; // G. C. Greubel, Oct 03 2024

Mathematica

b[n_, k_]:= Binomial[n-3, Floor[n/2]-k];
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]);
A110207[n_]:= If[n<3, f[n]/2, f[n]];
Table[A110207[n], {n,30}] (* G. C. Greubel, Oct 03 2024 *)

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)}

SageMath

def b(n,k): return binomial(n-3, (n//2) - k)
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))
[A110207(n) for n in range(1,31)] # G. C. Greubel, Oct 03 2024

Formula

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).