A193263 G.f.: A(x) = x + Sum_{n>=1} x^(2*n) / (1+x)^A193259(n).

1, 1, -1, 2, -5, 12, -26, 52, -101, 201, -422, 927, -2070, 4579, -9894, 20789, -42517, 84937, -166570, 322700, -622500, 1207056, -2376168, 4787523, -9908610, 21021499, -45404102, 98952388, -215756156, 467541948, -1002478352, 2121546013, -4427208709, 9110572776, -18503242145, 37135048484, -73759839074

Offset

1,4

Comments

Note: A193259(n) = n + floor(log_2(n)) + A011371(n), where A011371(n) = highest power of 2 dividing n!.

The g.f. A(x), as a power series in x, diverges at x=-1/2 and converges at x=+1/2 to A(1/2) = 0.6811907120229079095390167697...

Other values: A(x) = 1/2 at x = 0.385874434537804442263..., A(x) = 1 at x = 0.685568171776262105563..., A((sqrt(5)-1)/2) = 0.880771363850914609641...

Links

Table of n, a(n) for n=1..37.

Formula

G.f.: A(x) = Sum_{n>=1} x^n / (1+x)^(A193259(n) - n).

Example

G.f.: A(x) = x + x^2 - x^3 + 2*x^4 - 5*x^5 + 12*x^6 - 26*x^7 + 52*x^8 +...
where
A(x) = x + x^2/(1+x)^1 + x^4/(1+x)^4 + x^6/(1+x)^5 + x^8/(1+x)^9 + x^10/(1+x)^10 + x^12/(1+x)^12 + x^14/(1+x)^13 + x^16/(1+x)^18 +...+ x^(2*n)/(1+x)^A193259(n) +...
Also,
A(x) = x/(1+x)^0 + x^2/(1+x)^2 + x^3/(1+x)^2 + x^4/(1+x)^5 + x^5/(1+x)^5 + x^6/(1+x)^6 + x^7/(1+x)^6 + x^8/(1+x)^10 +...+ x^n/(1+x)^(A193259(n)-n) +...

Prog

(PARI) {a(n)=polcoeff(sum(m=1, n, x^m/(1+x+x*O(x^n))^(floor(log(m+1/2)/log(2)) + valuation(m!, 2))), n)}

Crossrefs

Cf. A193259, A011371, A000523.

Sequence in context: A221948 A116717 A116725 * A221949 A262803 A026622

Adjacent sequences: A193260 A193261 A193262 * A193264 A193265 A193266

Keyword

sign

Author

Paul D. Hanna, Jul 19 2011

Status

approved