OFFSET
0,4
COMMENTS
Conjecture: for n > 0, a(n) is odd iff n is a power of 2.
Conjecture: a(n) == 2 (mod 4) iff n/2 is a sum of two distinct powers of 2 (A018900).
a(n) ~ c*n^n where c appears to exist; c = 0.1469010938679... at n = 3000. What is an approximate value of this constant (if it exists and is finite)?
c = (exp(1) - 1)^2 / exp(3) = 0.1469959430660808807868671958665538372622476... - Vaclav Kotesovec, Mar 15 2026
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1024
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) [x^n] 1/(A(x) - n*x) = 0 for n >= 1.
(2) [x^n] 1/A(x) = -Sum_{k=1..n} n^k * [x^(n-k)] 1/A(x)^(k+1) for n >= 1.
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 31*x^4 + 348*x^5 + 5042*x^6 + 88688*x^7 + 1824547*x^8 + 42820188*x^9 + 1126268086*x^10 + ...
where [x^n] 1/(A(x) - n*x) = 0 for n >= 1.
ILLUSTRATION OF DEFINITION.
The table of coefficients of x^k in 1/(A(x) - n*x) begins
n = 0: [1,-1, 0, -3, -24, -290, -4368, -78799, ...];
n = 1: [1, 0, -1, -4, -30, -340, -4965, -87756, ...];
n = 2: [1, 1, 0, -5, -40, -414, -5744, -98745, ...];
n = 3: [1, 2, 3, 0, -42, -506, -6801, -112744, ...];
n = 4: [1, 3, 8, 17, 0, -490, -7872, -130251, ...];
n = 5: [1, 4, 15, 52, 146, 0, -7253, -145284, ...];
n = 6: [1, 5, 24, 111, 480, 1690, 0, -129301, ...];
n = 7: [1, 6, 35, 200, 1110, 5786, 24591, 0, ...];
n = 8: [1, 7, 48, 325, 2168, 14094, 86224, 430041, 0, ...]; ...
in which the main diagonal, the coefficient of x^n in row n, is all zeros after the initial term.
PROG
(PARI) {a(n) = my(A=[1], m); for(i=1, n, A = concat(A, 0); m=#A-1;
A[#A] = polcoef(1/(Ser(A) - m*x), m) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 05 2026
STATUS
approved