_Paul D. Hanna (pauldhanna(AT)juno.com), _, Oct 26 2007

a(n) = 2^[n(n+1) - A000120(n)] * [x^n] (1+x)^(1/2^n) for n>=0, where A000120(n) = number of 1's in binary expansion of n.

1, 1, -3, 35, -7285, 1570863, -2762459931, 9861642254451, -1141290059372782605, 66806775363324062981915, -31603810290612531279241668449, 30166547730607848261858185370275389, -464256425980552239880944863449968127087425

0,3

[x^n] (1+x)^(1/2^n) denotes the coefficient of x^n in the (2^n)-root of (1+x), which has a denominator equal to 2^[n(n+1) - A000120(n)].

This sequence forms the numerators of coefficients [x^n] (1+x)^(1/2^n),

where the denominators equal 2^b(n) and b(n) takes on values:

[0,1,5,10,19,28,40,53,71,88,108,129,154,179,207,236,271,304,...],

which is described by b(n) = n(n+1) - A000120(n) for n>=0.

(PARI) {a(n)=polcoeff((1+x+x*O(x^n))^(1/2^n), n)*2^(n*(n+1)-subst(Pol(binary(n)), x, 1))}

Cf. A000120; A134097 (variant); A134096.

sign,new

Paul D. Hanna (pauldhanna(AT)juno.com), Oct 26 2007

approved