OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1024
FORMULA
G.f. satisfies: A(x) = (1+x^2)/(1-x)^2 * A(x^2).
Define BISECTIONS: A(x) = B_0(x^2) + x*B_1(x^2), then: B_1(x)/B_0(x) = 2/(1+x).
EXAMPLE
G.f.: A(x) = 1 + 2*x + 6*x^2 + 10*x^3 + 22*x^4 + 34*x^5 + 62*x^6 +...
The g.f. satisfies:
A(x)/A(x^2) = 1 + 2*x + 4*x^2 + 6*x^3 + 8*x^4 + 10*x^5 +...+ 2*n*x^n +...
The logarithm of the g.f. begins:
log(A(x)) = 2*x + 8*x^2/2 + 2*x^3/3 + 16*x^4/4 + 2*x^5/5 + 8*x^6/6 + 2*x^7/7 + 32*x^8/8 + 2*x^9/9 + 8*x^10/10 + 2*x^11/11 + 16*x^12/12 +...+ 2^A090740(n)*x^n/n +...
where the highest exponents of 2 in 3^n-1, for n>=1, begins:
A090740 = [1,3,1,4,1,3,1,5,1,3,1,4,1,3,1,6,1,3,1,4,1,3,1,5,1,3,1,4,1,...].
The g.f.s of the BISECTIONS begin:
B_0(x) = 1 + 6*x + 22*x^2 + 62*x^3 + 150*x^4 + 326*x^5 + 654*x^6 +...
B_1(x) = 2 + 10*x + 34*x^2 + 90*x^3 + 210*x^4 + 442*x^5 + 866*x^6 +...
where 2 * B_0(x) / B_1(x) = 1+x.
PROG
(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, 2^valuation(3^m-1, 2)*x^m/m)+x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(i=1, #binary(n)+1, A=(1+x^2)/(1-x)^2*subst(A, x, x^2+x*O(x^n))); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 17 2012
STATUS
approved