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

1,2

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).

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)

Mathematica

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

Prog

(PARI) a(n)=polcoeff(x*(1+8*x)/((1-4*x)*(1-8*x)*(1-16*x)+x*O(x^n)), n)
(Magma) [Binomial(2^n, 2)^2: n in [1..30]]; // G. C. Greubel, Oct 02 2024
(SageMath)
def A110206(n): return binomial(2^n, 2)^2
[A110206(n) for n in range(1, 31)] # G. C. Greubel, Oct 02 2024

Crossrefs

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

Sequence in context: A004338 A228331 A014772 * A133051 A215605 A173187

Adjacent sequences: A110203 A110204 A110205 * A110207 A110208 A110209

Keyword

nonn

Author

Paul D. Hanna, Jul 16 2005

Status

approved