OFFSET
0,3
COMMENTS
Compare e.g.f. to: [x^(n-1)] exp(x)^n = [x^n] exp(x)^n for n>=1.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..160
FORMULA
The logarithm of the e.g.f. A(x) is an integer series:
_ log(A(x)) = Sum_{n>=1} A296175(n) * x^n.
E.g.f. A(x) satisfies:
_ 1/n! * d^n/dx^n A(x)^(n^4) = 1/(n-1)! * d^(n-1)/dx^(n-1) A(x)^(n^4) for n>=1, when evaluated at x = 0.
a(n) ~ -sqrt(1-c) * 2^(8*n - 8) * n^(4*n - 4) / (c^n * (4-c)^(3*n - 4) * exp(4*n)), where c = -LambertW(-4*exp(-4)) = 0.079309605127113656439108647386463779474372... - Vaclav Kotesovec, Oct 13 2020
EXAMPLE
E.g.f.: A(x) = 1 + x - 13*x^2/2! - 2999*x^3/3! - 8197751*x^4/4! - 81738176899*x^5/5! - 2105524335759389*x^6/6! - 115916378979693710123*x^7/7! - 12069952631345502122877199*x^8/8! - 2179911119857340269414590758951*x^9/9! - 639738016495616440994202167765715629*x^10/10! +...
To illustrate [x^(n-1)] A(x)^(n^4) = [x^n] A(x)^(n^4), form a table of coefficients of x^k in A(x)^(n^4) that begins as
n=1: [(1), (1), -13/2, -2999/6, -8197751/24, -81738176899/120, ...];
n=2: [1, (16), (16), -26992/3, -16767472/3, -164706495728/15, ...];
n=3: [1, 81, (5427/2), (5427/2), -246155517/8, -2300014714833/40, ...];
n=4: [1, 256, 30976, (6633728/3), (6633728/3), -2939838787328/15, ...];
n=5: [1, 625, 381875/2, 225885625/6, (122571375625/24), (122571375625/24), ...];
n=6: [1, 1296, 830736, 350400816, 108698540976, (126219948303024/5), (126219948303024/5), ...];
...
in which the diagonals indicated by parenthesis are equal.
Dividing the coefficients of x^(n-1)/(n-1)! in A(x)^(n^4) by n^4, we obtain the following sequence:
[1, 1, 67, 51826, 196114201, 2337406450056, 68145136372652611, 4136219111307043556272, 467591060765602023501093201, ...].
LOGARITHMIC PROPERTY.
Amazingly, the logarithm of the e.g.f. A(x) is an integer series:
log(A(x)) = x - 7*x^2 - 493*x^3 - 341101*x^4 - 680813601*x^5 - 2923660883625*x^6 - 22996362478599551*x^7 - 299331006952284448127*x^8 - 6006951481145880962408552*x^9 - 176288642409787912257773903552*x^10 - 7260231964238768891891716773249396*x^11 - 405879958110794676900559524931590299892*x^12 +...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^4)); A[#A] = (V[#A-1] - V[#A])/(#A-1)^4 ); n!*A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Dec 07 2017
STATUS
approved