OFFSET
0,4
COMMENTS
Conjecture: all terms are odd.
Conjecture: a(n) == 2 (mod 3) for n > 2.
Conjecture: a(n) == 3 (mod 4) for n > 2.
The limit lim_{n->oo} a(n)/n!^2 appears to be finite and nonzero. At n = 1000, a(1000)/1000!^2 = 0.576238922856...
This constant is equal to 0.5773928444790959583518657851093628462973104143119... - Vaclav Kotesovec, Jun 20 2026
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..500
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) n * [x^n] A(x)^n = (n-1) * [x^n] A(x)^(n+1) for n > 1.
(2) [x^n] A(x)^n = A397243(n) for n >= 1, where A397243 forms the logarithmic derivative of A397242.
(3) [x^n] A(x)^(n+1) = (n+1) * A397242(n) for n >= 0, where Sum_{n>=0} A397242(n)*x^n = exp(x + Sum_{n>=2} (n^2-1) * A397242(n)*x^n/n^2 ).
(4) A(x) = F(x*A(x)) where F(x) = A(x/F(x)) is the g.f. of A397242.
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 11*x^3 + 191*x^4 + 5279*x^5 + 205739*x^6 + 10703027*x^7 + 716263967*x^8 + 60007934687*x^9 + ...
The table of coefficients of x^k in A(x)^n begins
n=1: [1, 1, 1, 11, 191, 5279, 205739, 10703027, ...];
n=2: [1, 2, 3, 24, 405, 10962, 422539, 21832292, ...];
n=3: [1, 3, 6, 40, 645, 17085, 651040, 33405174, ...];
n=4: [1, 4, 10, 60, 915, 23688, 891932, 45439952, ...];
n=5: [1, 5, 15, 85, 1220, 30816, 1145960, 57955870, ...];
n=6: [1, 6, 21, 116, 1566, 38520, 1413930, 70973208, ...];
n=7: [1, 7, 28, 154, 1960, 46858, 1696716, 84513360, ...];
n=8: [1, 8, 36, 200, 2410, 55896, 1995268, 98598920, ...]; ...
from which we can illustrate the definition
n * [x^n] A(x)^n = (n-1) * [x^n] A(x)^(n+1) by
n = 2: 2*[x^2] A(x)^2 = 2*3 = 1*6 = 1*[x^2] A(x)^3;
n = 3: 3*[x^3] A(x)^3 = 3*40 = 2*60 = 2*[x^3] A(x)^4;
n = 4: 4*[x^4] A(x)^4 = 4*915 = 3*1220 = 3*[x^4] A(x)^5;
n = 5: 5*[x^5] A(x)^5 = 5*30816 = 4*38520 = 4*[x^5] A(x)^6;
etc.
Also, we obtain A397242 from the main diagonal
A397242 = [1/1, 2/2, 6/3, 60/4, 1220/5, 38520/6, ...]
= [1, 1, 2, 15, 244, 6420, 242388, 12324865, ...].
Further, we obtain A397243 from the secondary diagonal
A397243 = [1, 3, 40, 915, 30816, 1413930, ...]
which forms the logarithmic derivative of A397242.
PROG
(PARI) \\ returns vector of terms a(k) for k = 0..n
{a(n) = my(A=[1, 1], m); for(i=1, n-1, A=concat(A, 0); m=#A-1;
A[#A] = (m-1)*polcoef(Ser(A)^(m+1), m) - m*polcoef(Ser(A)^m, m) ); GF=Ser(A); A}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 19 2026
STATUS
approved