OFFSET
0,3
COMMENTS
It is remarkable that the logarithm of the e.g.f. A(x) should be an integer series.
Periodic modulo 10: a(5*n+k) = [1,1,5,5,5](k) (mod 10), for n>=0 and k = 0..4 (conjecture).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
a(n) ~ sqrt(1-c) * 3^(3*n - 7/3) * n^(3*n - 2) / (exp(3*n) * c^(n - 1/3) * (3-c)^(2*n - 2)), where c = -LambertW(-3*exp(-3)) = 0.1785606278779211065968086697... = -A226750. - Vaclav Kotesovec, Aug 07 2018
EXAMPLE
E.g.f.: A(x) = 1 + x + 5*x^2/2! + 445*x^3/3! + 196105*x^4/4! + 221673401*x^5/5! + 501981700621*x^6/6! + 1983064113021685*x^7/7! + 12488526496641458705*x^8/8! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp(n^3*x) / A(x)^(n^2) begins:
n=1: [1, 0, -4, -432, -194256, -220662720, -500627544000, ...];
n=2: [1, 4, 0, -1856, -805376, -898258176, -2023715201024, ...];
n=3: [1, 18, 288, 0, -1989792, -2154563712, -4727980751616, ...];
n=4: [1, 48, 2240, 94464, 0, -4244861952, -9137589559296, ...];
n=5: [1, 100, 9900, 959200, 84852400, 0, -15901448888000, ...];
n=6: [1, 180, 32256, 5738688, 1003636224, 161358324480, 0, ...];
n=7: [1, 294, 86240, 25218144, 7335234144, 2103824749824, 557359956846336, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 2*x^2 + 72*x^3 + 8096*x^4 + 1839000*x^5 + 695334816*x^6 + 392764566208*x^7 + 309340607492096*x^8 + ... + A317346(n)*x^n + ...
PROG
(PARI) {a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^3*x +x*O(x^#A)) / Ser(A)^(m^2) )[m+1]/m^2 ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 26 2018
STATUS
approved