A334921 - OEIS

%I #9 May 23 2020 05:28:46

%S 1,-2,2,0,-6,20,-48,96,-166,252,-340,416,-480,544,-544,0,2906,-13396,

%T 44100,-121792,296860,-652808,1306560,-2377280,3879136,-5461952,

%U 5892512,-2171520,-11699616,45871040,-114213888,228427776,-377994406,478195212,-252252460,-1013309824

%N Expansion of Phi(x) = (1/(1+x))*Product_{k>=0} (1-(x/(1+x))^2^k).

%C The Hankel transforms of Phi(x) and Phi(x^2) are identical. See Theorem 2.8 in Han paper.

%H Michel Marcus, <a href="/A334921/b334921.txt">Table of n, a(n) for n = 0..1000</a>

%H Guo-Niu Han, <a href="https://hal.archives-ouvertes.fr/hal-02125293">Jacobi continued fraction and Hankel determinants of the Thue-Morse sequence</a>, Quaestiones Mathematicae, 2016, 39 (7), pp.895-909. hal-02125285.

%F a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*(-1)^(2*n-2*k+A000120(k))). See Theorem 2.8 in Han paper.

%F a(n) = 0 for positive terms of A024036.

%t a[n_] := Sum[(-1)^(n - k) * Binomial[n, k] * (-1)^(2*n - 2*k + DigitCount[k, 2, 1]), {k, 0, n}]; Array[a, 36, 0] (* _Amiram Eldar_, May 16 2020 *)

%o (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*(-1)^(2*n-2*k+hammingweight(k)));

%Y Cf. A000120 (Hamming weight of n), A024036 (4^n - 1).

%K sign

%O 0,2

%A _Michel Marcus_, May 16 2020