OFFSET
0,4
COMMENTS
Compare to: Sum{n>=0} Series_Reversion( x/(1 + x^n)^(1/n) )^(n^2) = Sum_{n>=0} x^(n^2)/(1 - x^n)^n, the g.f. of A143862.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..400
FORMULA
G.f.: Sum{n>=0} (1 - sqrt(1 - 4*x^n))^n / 2^n.
G.f.: Sum{n>=0} Series_Reversion( x*(1 - x^n)^(1/n) )^(n^2).
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 6*x^4 + 14*x^5 + 44*x^6 + 132*x^7 + 434*x^8 + 1431*x^9 + 4876*x^10 + 16796*x^11 + 58831*x^12 + ...
such that
A(x) = 1 + (1 - sqrt(1 - 4*x))/2 + (1 - sqrt(1 - 4*x^2))^2/2^2 + (1 - sqrt(1 - 4*x^3))^3/2^3 + (1 - sqrt(1 - 4*x^4))^4/2^4 + (1 - sqrt(1 - 4*x^5))^5/2^5 + (1 - sqrt(1 - 4*x^6))^6/2^6 + ...
The related series x^(n^2) * C(x^n)^n = (1 - sqrt(1 - 4*x^n))^n/2^n begin:
n=1: x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + ...;
n=2: x^4 + 2*x^6 + 5*x^8 + 14*x^10 + 42*x^12 + ...;
n=3: x^9 + 3*x^12 + 9*x^15 + 28*x^18 + 90*x^21 + ...;
n=4: x^16 + 4*x^20 + 14*x^24 + 48*x^28 + 165*x^32 + ...;
n=5: x^25 + 5*x^30 + 20*x^35 + 75*x^40 + 275*x^45 + ...;
n=6: x^36 + 6*x^42 + 27*x^48 + 110*x^54 + 429*x^60 + ...;
...
SPECIFIC VALUES.
A(1/4) = Sum_{n>=0} (2^(n-1) - sqrt(4^(n-1) - 1))^n / 2^(n^2) = 1.504491300666... = 1 + 1/2 + (2 - sqrt(3))^2/2^4 + (4 - sqrt(15))^3/2^9 + (8 - sqrt(63))^4/2^16 + (16 - sqrt(255))^5/2^25 + (32 - sqrt(1023))^6/2^36 + (64 - sqrt(4095))^7/2^49 + ...
A(-1/4) = Sum_{n>=0} (2^(n-1) - sqrt(4^(n-1) + 1))^n / 2^(n^2) = 0.79637258079... = 1 + (1 - sqrt(2))/2 + (2 - sqrt(5))^2/2^4 + (4 - sqrt(17))^3/2^9 + (8 - sqrt(65))^4/2^16 + (16 - sqrt(257))^5/2^25 + (32 - sqrt(1025))^6/2^36 + ...
PROG
(PARI) {a(n) = my(A); A = sum(m=0, sqrtint(n+1), (1 - sqrt(1 - 4*x^m +x*O(x^n) ))^m / 2^m); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 16 2018
STATUS
approved