OFFSET
0,2
COMMENTS
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f.: Sum_{n>=0} (1+x)^(n^2) * Sum_{k=0..n} (-1)^k * binomial(n,k) / (1+x)^(2*n*k).
G.f.: Sum_{n>=0} 1/(1+x)^(n^2) * Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * (1+x)^(2*n*k).
a(n) ~ c * d^n * n! / sqrt(n), where d = 5.4666049332127684665699843922982444983683628264382802770893... and c = 0.3563391278539240852770166562386253680399190992740998... - Vaclav Kotesovec, Oct 10 2020
EXAMPLE
G.f.: A(x) = 1 + 2*x + 15*x^2 + 201*x^3 + 3807*x^4 + 93103*x^5 + 2788528*x^6 + 98816388*x^7 + 4043274742*x^8 + 187583369889*x^9 + ...
such that
A(x) = 1 + ((1+x) - 1/(1+x)) + ((1+x)^2 - 1/(1+x)^2)^2 + ((1+x)^3 - 1/(1+x)^3)^3 + ((1+x)^4 - 1/(1+x)^4)^4 + ((1+x)^5 - 1/(1+x)^5)^5 + ...
Equivalently,
A(x) = 1 +
((1+x) - 1/(1+x)) +
((1+x)^4 - 2 + 1/(1+x)^4) +
((1+x)^9 - 3*(1+x)^3 + 3/(1+x)^3 - 1/(1+x)^9) +
((1+x)^16 - 4*(1+x)^8 + 6 - 4/(1+x)^8 + 1/(1+x)^16) +
((1+x)^25 - 5*(1+x)^15 + 10*(1+x)^5 - 10/(1+x)^5 + 5/(1+x)^15 - 1/(1+x)^25) +
((1+x)^36 - 6*(1+x)^24 + 15*(1+x)^12 - 20 + 15/(1+x)^12 - 6/(1+x)^24 + 1/(1+x)^36) + ...
PROG
(PARI) {a(n) = my(A=1, X=x + x*O(x^n)); A = sum(m=0, n, ((1+x)^m - 1/(1+X)^m)^m ); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 28 2018
STATUS
approved