OFFSET
1,2
COMMENTS
It is remarkable that this sequence should consist entirely of integers.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..200
FORMULA
a(n) ~ sqrt(1-c) * 2^(8*n - 7) * n^(3*n - 7/2) / (sqrt(Pi) * c^(n - 1/4) * (4-c)^(3*n - 3) * exp(3*n)), where c = -LambertW(-4*exp(-4)) = 0.079309605127113656439108647386463779474372628792... - Vaclav Kotesovec, Oct 13 2020
EXAMPLE
G.f.: A(x) = x + 4*x^2 + 756*x^3 + 666368*x^4 + 1573194000*x^5 + 7732870168896*x^6 + 68447889079649744*x^7 + 991668063561584680960*x^8 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^4*x - n^3*A(x) ) begins:
n=1: [1, 0, -8, -4536, -15992640, -188782917120, ...];
n=2: [1, 8, 0, -37312, -129112064, -1515383749632, ...];
n=3: [1, 54, 2700, 0, -453396528, -5216886215136, ...];
n=4: [1, 192, 36352, 6492672, 0, -12944825376768, ...];
n=5: [1, 500, 249000, 122933000, 57869896000, 0, ...];
n=6: [1, 1080, 1164672, 1253133504, 1340717598720, 1376766221921280, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 9*x^2/2! + 4561*x^3/3! + 16011217*x^4/4! + 188863653441*x^5/5! + 5568801588349081*x^6/6! + 345016372861620194449*x^7/7! + ...
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^4*x +x*O(x^#A)) / Ser(A)^(m^3) )[m+1]/m^3 ); polcoeff( log(Ser(A)), n)}
for(n=1, 15, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 02 2018
STATUS
approved