OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..400
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) [x^n] A(x)^(n+1) * (1 - (n+1)^2*x*A(x)) = 0 for n >= 1.
(2) A(x) = 1 + x*A(x)^3 * (2*A(x)^2 - 2*x^2*A'(x)^2 + x^2*A(x)*A''(x)) / (A(x) - x*A'(x))^3.
(3) A(x) = x/Series_Reversion(G(x)) where G(x) = x + x*d^2/dx^2 G^3/3 is the g.f. of A393751, so that G(x) = x*A(G(x)).
a(n) ~ c * n!^2 * n^3, where c = 0.540738861384411785634541505306707722567... - Vaclav Kotesovec, Mar 15 2026
EXAMPLE
G.f.: A(x) = 1 + 2*x + 20*x^2 + 432*x^3 + 15008*x^4 + 732032*x^5 + 46719936*x^6 + 3746127360*x^7 + 367655264768*x^8 + ...
The table of coefficients of x^k in A(x)^(n+1)*(1 - (n+1)^2*x*A(x)) begins
n = 0: [1, 1, 16, 388, 14064, 699888, ...];
n = 1: [1, 0, 20, 656, 25968, 1334784, ...];
n = 2: [1, -3, 0, 608, 31488, 1771632, ...];
n = 3: [1, -8, -56, 0, 25136, 1842176, ...];
n = 4: [1, -15, -160, -1460, 0, 1337152, ...];
n = 5: [1, -24, -324, -4112, -52416, 0, ...]; ...
in which the main diagonal, the coefficient of x^n in row n, is all zeros after the initial term.
RELATED SERIES.
Series G(x) = x + x*d^2/dx^2 G^3/3 is the g.f. of A393751, which begins
G(x) = x + 2*x^2 + 24*x^3 + 560*x^4 + 19760*x^5 + 952224*x^6 + 59446016*x^7 + ... + A393751(n)*x^n + ...
where G(x) = x*A(G(x)).
Other properties of G(x) include
(i) G(x/A(x)) = x,
(ii) G'(x/A(x)) = A(x)^2/(A(x) - x*A'(x)),
(iii) G''(x/A(x)) = A(x)^3 * (2*A(x)*A'(x) - 2*x*A'(x)^2 + x*A(x)*A''(x)) / (A(x) - x*A'(x))^3,
(iv) G(x) = x + 2*x*G(x)*G'(x)^2 + x*G(x)^2*G''(x).
If we substitute x/A(x) for x in (iv) then simplify by using (i)-(iii), we obtain the formula A(x) = 1 + x*A(x)^3 * (2*A(x)^2 - 2*x^2*A'(x)^2 + x^2*A(x)*A''(x)) / (A(x) - x*A'(x))^3.
PROG
(PARI) \\ From [x^n] A(x)^(n+1) * (1 - (n+1)^2*A(x)) = 0:
{a(n) = my(V=[1], A, m); for(i=0, n, V=concat(V, 0); A = Ser(V); m=#V;
V[m] = -Vec( A^m*(1 - m^2*x*A))[m]/m); polcoef(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) \\ From differential equation
{a(n) = my(A=1+x); for(i=1, n, A = 1 + x*A^3*(2*A^2 - 2*x^2*A'^2 + x^2*A*A'')/(A - x*A' +x*O(x^n))^3 ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 07 2026
STATUS
approved