A110206 Row sums of triangle A110205, where A110205(n,k) equals the sum of cubes of numbers < 2^n having exactly k ones in their binary expansion.

1, 36, 784, 14400, 246016, 4064256, 66064384, 1065369600, 17112825856, 274341298176, 4393752592384, 70334388633600, 1125625045712896, 18012199553335296, 288212784234102784, 4611545282012774400

Offset: 1

Author

Paul D. Hanna, Jul 16 2005

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..820
• Index entries for linear recurrences with constant coefficients, signature (28,-224,512).

Crossrefs

Cf. A110205 (triangle), A110207 (central terms).

Programs

Magma

[Binomial(2^n,2)^2: n in [1..30]]; // G. C. Greubel, Oct 02 2024

Mathematica

Binomial[2^Range[30], 2]^2 (* G. C. Greubel, Oct 02 2024 *)

PARI

a(n)=polcoeff(x*(1+8*x)/((1-4*x)*(1-8*x)*(1-16*x)+x*O(x^n)),n)

SageMath

def A110206(n): return binomial(2^n, 2)^2
[A110206(n) for n in range(1,31)] # G. C. Greubel, Oct 02 2024

Formula

G.f.: x*(1+8*x)/( (1-4*x)*(1-8*x)*(1-16*x) ).

From G. C. Greubel, Oct 02 2024: (Start)

a(n) = ( binomial(2^n, 2) )^2 = 4^(n-1)*(2^n - 1)^2.

E.g.f.: (1/4)*(exp(4*x) - 2*exp(8*x) + exp(16*x)). (End)