OFFSET
0,3
COMMENTS
Compare e.g.f. to: [x^n] exp(x)^(n^2) = n * [x^(n-1)] exp(x)^(n^2) for n>=1.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
E.g.f. A(x) satisfies: log(A(x)) = Sum_{n>=1} A300593(n)*x^n, a power series in x with integer coefficients.
a(n) ~ c * n!^4, where c = 3.1056678107899395562612789210816... - Vaclav Kotesovec, Oct 14 2020
EXAMPLE
E.g.f.: A(x) = 1 + x + 13*x^2/2! + 1333*x^3/3! + 438073*x^4/4! + 328561681*x^5/5! + 482408372341*x^6/6! + 1262989939509733*x^7/7! + 5507311107090685873*x^8/8! + 37883505322347710775553*x^9/9! + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in A(x)^(n^2) begins:
n=1: [(1), (1), 13/2, 1333/6, 438073/24, 328561681/120, ...];
n=2: [1, (4), (32), 2912/3, 228032/3, 167874308/15, ...];
n=3: [1, 9, (189/2), (5103/2), 1468467/8, 1045214163/40, ...];
n=4: [1, 16, 224, (17024/3), (1089536/3), 735471632/15, ...];
n=5: [1, 25, 925/2, 70525/6, (15835225/24), (1979403125/24), ...];
n=6: [1, 36, 864, 23328, 1161792, (654796044/5), (141435945504/5), ...]; ...
in which the coefficients in parenthesis are related by
1 = 1*1; 32 = 2^3*4; 5103/2 = 3^3*189/2; 1089536/3 = 4^3*17024/3; ...
illustrating that: [x^n] A(x)^(n^2) = n^3 * [x^(n-1)] A(x)^(n^2).
LOGARITHMIC PROPERTY.
The logarithm of the e.g.f. is the integer series:
log(A(x)) = x + 6*x^2 + 216*x^3 + 18016*x^4 + 2718575*x^5 + 667151244*x^6 + 249904389518*x^7 + 136335045655680*x^8 + 104258627494173747*x^9 + 108236370325030253850*x^10 + 148475074256982964816314*x^11 + 263023328027145941803648512*x^12 + ... + A300593(n)*x^n + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n+1, A=concat(A, 0); V=Vec(Ser(A)^((#A-1)^2)); A[#A] = ((#A-1)^3*V[#A-1] - V[#A])/(#A-1)^2 ); n!*A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 09 2018
STATUS
approved