OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..340
FORMULA
G.f.: Sum_{n>=0} (1+x)^(2*n^2) /(1 + (1+x)^(2*n))^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = 2*A317855 = 6.3221773077308576276603444051762649834527655483771126832545564150753941184386... and c = 0.302715376391132275494451399946850989516917... - Vaclav Kotesovec, Aug 09 2018
EXAMPLE
G.f.: A(x) = 1 + 2*x + 17*x^2 + 264*x^3 + 5784*x^4 + 163610*x^5 + 5667551*x^6 + 232280480*x^7 + 10991951114*x^8 + 589780778314*x^9 + ...
such that
A(x) = 1 + ((1+x)^2-1) + ((1+x)^4-1)^2 + ((1+x)^6-1)^3 + ((1+x)^8-1)^4 + ((1+x)^10-1)^5 + ((1+x)^12-1)^6 + ((1+x)^14-1)^7 + ...
Also,
A(x) = 1/2 + (1+x)^2/(1 + (1+x)^2)^2 + (1+x)^8/(1 + (1+x)^4)^3 + (1+x)^18/(1 + (1+x)^6)^4 + (1+x)^32/(1 + (1+x)^8)^5 + (1+x)^50/(1 + (1+x)^10)^6 + ...
PROG
(PARI) {a(n) = my(A, o=x*O(x^n)); A = sum(m=0, n, ((1+x +o)^(2*m) - 1)^m ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 24 2018
STATUS
approved