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

1,2

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

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).

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 *)

Prog

(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)}
(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
(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

Crossrefs

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

Sequence in context: A223278 A278091 A042410 * A223205 A257786 A161530

Adjacent sequences: A110204 A110205 A110206 * A110208 A110209 A110210

Keyword

nonn,changed

Author

Paul D. Hanna, Jul 16 2005

Status

approved