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A385082 Sum of squared coefficients of Product_{i=0..n-1} 1+x^(2^i+1)+x^(2^(i+1)+1).

%I A385082 #17 Jun 18 2025 16:02:11
%S A385082 1,3,13,55,249,1121,5025,22607,101931,460877,2088687,9482763,43109307,
%T A385082 196163983,893222041,4069162197,18543631161,84525140297,385343891847,
%U A385082 1756959373157,8011450183181,36533108258455,166602342944307,759783053580809,3465042771956289,15802856371611411
%N A385082 Sum of squared coefficients of Product_{i=0..n-1} 1+x^(2^i+1)+x^(2^(i+1)+1).
%H A385082 Shalosh B. Ekhad and Doron Zeilberger, <a href="https://arxiv.org/abs/2103.12855">Automated Generation of Generating Functions Related to Generalized Stern's Diatomic Arrays in the footsteps of Richard Stanley</a>, arXiv:2103.12855 [math.CO], 2021-2024.
%H A385082 Jinlong Tang and Guoce Xin, <a href="https://arxiv.org/abs/2506.13375">Meeting a Challenge raised by Ekhad and Zeilberger related to Stern's Triangle</a>, arXiv:2506.13375 [math.CO], 2025.
%p A385082 b:= proc(n) option remember; expand(`if`(n<0, 1,
%p A385082        b(n-1)*(1+x^(2^n+1)+x^(2^(n+1)+1))))
%p A385082     end:
%p A385082 a:= n-> add(i^2, i=[coeffs(b(n-1))]):
%p A385082 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jun 17 2025
%t A385082 a[n_]:=Total[CoefficientList[Product[ 1+x^(2^i+1)+x^(2^(i+1)+1),{i,0,n-1}],x]^2]; Array[a,20,0] (* _Stefano Spezia_, Jun 17 2025 *)
%o A385082 (PARI) a(n) = norml2(Vec(prod(i=0, n-1, 1+x^(2^i+1)+x^(2^(i+1)+1))));
%o A385082 (Python)
%o A385082 from collections import Counter
%o A385082 from itertools import count, islice
%o A385082 def A385082_gen(): # generator of terms
%o A385082     c = Counter({0:1})
%o A385082     for n in count(0):
%o A385082         yield sum(i**2 for i in c.values())
%o A385082         c = sum((Counter({i:j,(m:=1<<n)+i+1:j,(m<<1)+i+1:j}) for i, j in c.items()),start=Counter())
%o A385082 A385082_list = list(islice(A385082_gen(),10)) # _Chai Wah Wu_, Jun 18 2025
%Y A385082 Cf. A052984 (with 1+x^(2^i)+x^(2^(i+1)) instead).
%K A385082 nonn
%O A385082 0,2
%A A385082 _Michel Marcus_, Jun 16 2025