A393750 - OEIS

%I #19 Mar 15 2026 11:40:16

%S 1,2,20,432,15008,732032,46719936,3746127360,367655264768,

%T 43351846829568,6053258934248448,988875035437678592,

%U 187032270760583086080,40577940457446867402752,10015352742515991069949952,2791377985885263046890749952,872656014865883705124278894592,304171471169773991468448233553920,117563365979571447840991174074040320

%N G.f. A(x) satisfies [x^n] A(x)^(n+1) * (1 - (n+1)^2*x*A(x)) = 0 for n >= 1.

%H Paul D. Hanna, <a href="/A393750/b393750.txt">Table of n, a(n) for n = 0..400</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

%F (1) [x^n] A(x)^(n+1) * (1 - (n+1)^2*x*A(x)) = 0 for n >= 1.

%F (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.

%F (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)).

%F a(n) ~ c * n!^2 * n^3, where c = 0.540738861384411785634541505306707722567... - _Vaclav Kotesovec_, Mar 15 2026

%e 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 + ...

%e The table of coefficients of x^k in A(x)^(n+1)*(1 - (n+1)^2*x*A(x)) begins

%e   n = 0: [1,   1,   16,   388,  14064,  699888, ...];

%e   n = 1: [1,   0,   20,   656,  25968, 1334784, ...];

%e   n = 2: [1,  -3,    0,   608,  31488, 1771632, ...];

%e   n = 3: [1,  -8,  -56,     0,  25136, 1842176, ...];

%e   n = 4: [1, -15, -160, -1460,      0, 1337152, ...];

%e   n = 5: [1, -24, -324, -4112, -52416,       0, ...]; ...

%e in which the main diagonal, the coefficient of x^n in row n, is all zeros after the initial term.

%e RELATED SERIES.

%e Series G(x) = x + x*d^2/dx^2 G^3/3 is the g.f. of A393751, which begins

%e 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 + ...

%e where G(x) = x*A(G(x)).

%e Other properties of G(x) include

%e (i) G(x/A(x)) = x,

%e (ii) G'(x/A(x)) = A(x)^2/(A(x) - x*A'(x)),

%e (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,

%e (iv) G(x) = x + 2*x*G(x)*G'(x)^2 + x*G(x)^2*G''(x).

%e 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.

%o (PARI) \\ From [x^n] A(x)^(n+1) * (1 - (n+1)^2*A(x)) = 0:

%o {a(n) = my(V=[1],A,m); for(i=0, n, V=concat(V, 0); A = Ser(V); m=#V;

%o V[m] = -Vec( A^m*(1 - m^2*x*A))[m]/m); polcoef(A,n)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) \\ From differential equation

%o {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)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A393751, A245118.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 07 2026