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 - 5/2) * n^(3*n - 3/2) / (sqrt(Pi) * exp(3*n) * c^n * (4-c)^(3*n - 1)), where c = -LambertW(-4*exp(-4)) = 0.079309605127113656439108647... - Vaclav Kotesovec, Oct 13 2020
B(x) = x + (1/(m+1)) * A((m+1)^4 * x^(m+1)) satisfies [x^n] exp(n*x*(1+n^3*x^m) - n*B(x) ) = 0 for n >= 1. - Seiichi Manyama, Apr 09 2026
EXAMPLE
G.f.: A(x) = x + 49*x^2 + 22542*x^3 + 34776266*x^4 + 124857847020*x^5 + 863035137487572*x^6 + 10208133235178252640*x^7 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k/k! in exp( n^4*x - n*A(x) ) begins:
n=1: [1, 0, -98, -135252, -834601572, -14982809095440, ...];
n=2: [1, 14, 0, -275992, -1684485824, -30082728311616, ...];
n=3: [1, 78, 5790, 0, -2603944836, -45947242627272, ...];
n=4: [1, 252, 63112, 15165648, 0, -63525640595328, ...];
n=5: [1, 620, 383910, 236740340, 140783667580, 0, ...];
n=6: [1, 1290, 1663512, 2143601928, 2754163718208, 3423991878509760, 0, ...]; ...
in which the coefficient of x^n in row n forms a diagonal of zeros.
RELATED SERIES.
exp(A(x)) = 1 + x + 99*x^2/2! + 135547*x^3/3! + 835200793*x^4/4! + 14987248838841*x^5/5! + 621476619810599851*x^6/6! + ...
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 )[m+1]/m ); polcoeff( log(Ser(A)), n)}
for(n=1, 15, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 02 2018
STATUS
approved