OFFSET
1,2
COMMENTS
E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! equals the logarithm of the e.g.f. of A319147.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..100
FORMULA
a(n) ~ c * d^n * n! / n^(5/2), where d = 6.1601834100761946... (same as for A319147) and c = 0.193396776821391327... - Vaclav Kotesovec, Mar 19 2024
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 184*x^4/4! + 3325*x^5/5! + 79056*x^6/6! + 2345539*x^7/7! + 83505920*x^8/8! + 3472829721*x^9/9! + ...
Exponentiation yields the e.g.f. of A319147:
exp(A(x)) = 1 + x + 3*x^2/2! + 22*x^3/3! + 269*x^4/4! + 4776*x^5/5! + 111967*x^6/6! + 3280264*x^7/7! + 115550073*x^8/8! +...+ A319147(n)*x^n/n! + ...
which equals
Limit_{N->oo} [ Sum_{n>=0} (N^2 + N*n + n^2)^n * (x/N)^n/n! ]^(1/N).
RELATED SEQUENCES.
a(n) is divisible by n where a(n)/n begins:
[1, 1, 5, 46, 665, 13176, 335077, 10438240, 385869969, 16532139520, ...].
PROG
(PARI) {a(n) = n! * polcoeff( polcoeff( log( sum(m=0, 2*n, (m^2 + m*y + y^2)^m *x^m/m! ) +x*O(x^(2*n)) ), n, x), n+1, y)}
for(n=1, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 30 2018
STATUS
approved