OFFSET
0,2
COMMENTS
The following identities hold for |y| <= 1 and fixed real k > 0:
(C1) Sum_{n>=0} (y^n + k)^n/(1+k + y^n)^(n+1) = Sum_{n>=0} (y^n - 1)^n/(1+k - k*y^n)^(n+1).
(C2) Sum_{n>=0} (y^n + 1)^n*k^n/(1+k + k*y^n)^(n+1) = Sum_{n>=0} (y^n - 1)^n*k^n/(1+k - k*y^n)^(n+1).
This sequence is an example of (C2) when y = 1+x and k = 3.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} ( (1+x)^n - 1 )^n * 3^n / (4 - 3*(1+x)^n)^(n+1).
(2) A(x) = Sum_{n>=0} ( (1+x)^n + 1 )^n * 3^n / (4 + 3*(1+x)^n)^(n+1).
a(n) ~ c * d^n * n! / sqrt(n), where d = 11.154788564351081167494585241180262193438722530344791058752757035461192417... and c = 0.321897864665202841967234839159770976446040882710871129852558... - Vaclav Kotesovec, Aug 09 2018
EXAMPLE
G.f.: A(x) = 1 + 3*x + 54*x^2 + 1494*x^3 + 58248*x^4 + 2921346*x^5 + 179119836*x^6 + 12981530772*x^7 + 1085678924472*x^8 + ...
such that
A(x) = 1 + ((1+x) - 1)*3/(4 - 3*(1+x))^2 + ((1+x)^2 - 1)^2*3^2/(4 - 3*(1+x)^2)^3 + ((1+x)^3 - 1)^3*3^3/(4 - 3*(1+x)^3)^4 + ((1+x)^4 - 1)^4*3^4/(4 - 3*(1+x)^4)^5 + ((1+x)^5 - 1)^5*3^5/(4 - 3*(1+x)^5)^6 + ((1+x)^6 - 1)^6*3^6/(4 - 3*(1+x)^6)^7 + ...
Also,
A(x) = 1/7 + ((1+x) + 1)*3/(4 + 3*(1+x))^2 + ((1+x)^2 + 1)^2*3^2/(4 + 3*(1+x)^2)^3 + ((1+x)^3 + 1)^3*3^3/(4 + 3*(1+x)^3)^4 + ((1+x)^4 + 1)^4*3^4/(4 + 3*(1+x)^4)^5 + ((1+x)^5 + 1)^5*3^5/(4 + 3*(1+x)^5)^6 + ((1+x)^6 + 1)^6*3^6/(4 + 3*(1+x)^6)^7 + ...
PROG
(PARI) {a(n) = my(A=1); A = sum(m=0, n, ( (1+x)^m - 1 +x*O(x^n) )^m * 3^m / (4 - 3*(1+x)^m +x*O(x^n) )^(m+1) ); ; polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 03 2018
STATUS
approved