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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.

%I A110206 #10 Oct 03 2024 08:31:50
%S A110206 1,36,784,14400,246016,4064256,66064384,1065369600,17112825856,
%T A110206 274341298176,4393752592384,70334388633600,1125625045712896,
%U A110206 18012199553335296,288212784234102784,4611545282012774400
%N 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.
%H A110206 G. C. Greubel, <a href="/A110206/b110206.txt">Table of n, a(n) for n = 1..820</a>
%H A110206 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (28,-224,512).
%F A110206 G.f.: x*(1+8*x)/( (1-4*x)*(1-8*x)*(1-16*x) ).
%F A110206 From _G. C. Greubel_, Oct 02 2024: (Start)
%F A110206 a(n) = ( binomial(2^n, 2) )^2 = 4^(n-1)*(2^n - 1)^2.
%F A110206 E.g.f.: (1/4)*(exp(4*x) - 2*exp(8*x) + exp(16*x)). (End)
%t A110206 Binomial[2^Range[30], 2]^2 (* _G. C. Greubel_, Oct 02 2024 *)
%o A110206 (PARI) a(n)=polcoeff(x*(1+8*x)/((1-4*x)*(1-8*x)*(1-16*x)+x*O(x^n)),n)
%o A110206 (Magma) [Binomial(2^n,2)^2: n in [1..30]]; // _G. C. Greubel_, Oct 02 2024
%o A110206 (SageMath)
%o A110206 def A110206(n): return binomial(2^n, 2)^2
%o A110206 [A110206(n) for n in range(1,31)] # _G. C. Greubel_, Oct 02 2024
%Y A110206 Cf. A110205 (triangle), A110207 (central terms).
%K A110206 nonn
%O A110206 1,2
%A A110206 _Paul D. Hanna_, Jul 16 2005