OFFSET
1,2
COMMENTS
E.g.f. A(x) = Sum_{n>=1} a(n)*x^(2*n-1)/(2*n-1)! equals the logarithm of the e.g.f. of A318633.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..145 (terms 1..75 from Paul D. Hanna)
FORMULA
a(n) ~ 5^(-1/4) * 2^(3*n - 3/2) * (1 + sqrt(5))^(n - 3/2) * exp((1 - sqrt(5))*n + (sqrt(5) - 3)/2) * n^(2*n-3). - Vaclav Kotesovec, Mar 20 2024
EXAMPLE
E.g.f.: A(x) = x + 6*x^3/3! + 480*x^5/5! + 122640*x^7/7! + 66044160*x^9/9! + 61482516480*x^11/11! + 88135315107840*x^13/13! + 180378921026304000*x^15/15! + ...
The e.g.f. A(x) may also be written using somewhat reduced coefficients
A(x) = x + x^3 + 8*x^5/2! + 146*x^7/3! + 4368*x^9/4! + 184832*x^11/5! + 10190656*x^13/6! + 695211120*x^15/7! + 56648897024*x^17/8! + 5374487515904*x^19/9! + ... + a(n)*(n-1)!/(2*n-1)! * x^(2*n-1)/(n-1)! + ...
Exponentiation yields the e.g.f. of A318633:
exp(A(x)) = 1 + x + x^2/2! + 7*x^3/3! + 25*x^4/4! + 541*x^5/5! + 3361*x^6/6! + 135451*x^7/7! + 1179697*x^8/8! + 72062425*x^9/9! +...+ A318633(n)*x^n/n! + ...
which equals
Limit_{N->oo} [ Sum_{n>=0} (N^2 + n^2)^n * (x/N)^n/n! ]^(1/N).
PROG
(PARI) {a(n) = (2*n-1)! * polcoeff( polcoeff( log( sum(m=0, 2*n, (m^2 + y^2)^m *x^m/m! ) +x*O(x^(2*n)) ), 2*n-1, x), 2*n, y)}
for(n=1, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 04 2018
STATUS
approved