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

1, 3, 13, 55, 249, 1121, 5025, 22607, 101931, 460877, 2088687, 9482763, 43109307, 196163983, 893222041, 4069162197, 18543631161, 84525140297, 385343891847, 1756959373157, 8011450183181, 36533108258455, 166602342944307, 759783053580809, 3465042771956289, 15802856371611411

Offset: 0

Author

Michel Marcus, Jun 16 2025

Keywords

• nonn

Links

• Shalosh B. Ekhad and Doron Zeilberger, Automated Generation of Generating Functions Related to Generalized Stern's Diatomic Arrays in the footsteps of Richard Stanley, arXiv:2103.12855 [math.CO], 2021-2024.
• Jinlong Tang and Guoce Xin, Meeting a Challenge raised by Ekhad and Zeilberger related to Stern's Triangle, arXiv:2506.13375 [math.CO], 2025.

Crossrefs

Cf. A052984 (with 1+x^(2^i)+x^(2^(i+1)) instead).

Programs

Maple

b:= proc(n) option remember; expand(`if`(n<0, 1,
       b(n-1)*(1+x^(2^n+1)+x^(2^(n+1)+1))))
    end:
a:= n-> add(i^2, i=[coeffs(b(n-1))]):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 17 2025

Mathematica

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 *)

PARI

a(n) = norml2(Vec(prod(i=0, n-1, 1+x^(2^i+1)+x^(2^(i+1)+1))));

Python

from collections import Counter
from itertools import count, islice
def A385082_gen(): # generator of terms
    c = Counter({0:1})
    for n in count(0):
        yield sum(i**2 for i in c.values())
        c = sum((Counter({i:j,(m:=1<A385082_list = list(islice(A385082_gen(),10)) # Chai Wah Wu, Jun 18 2025