A397132 E.g.f. A(x) satisfies A'(x) = (1 + A(x)*A'(x))^4, with A(0)=0.

1, 4, 60, 1608, 62520, 3195360, 202685760, 15367129920, 1356166333440, 136594585094400, 15465238193433600, 1944635519232806400, 268912358164611532800, 40563396480486872678400, 6628655092235188446720000, 1166636976398869777072128000, 220020666451278727461672960000

Offset

1,2

Comments

In general, for k > 1, if e.g.f. A(x) satisfies A'(x) = (1 + A(x)*A'(x))^k, then a(n) ~ k^(1/2) * ((4*k-2)/(1 + (1 - 1/k)^(2*k)))^(n - 3/2) * n^(n-2) / ((1 - 1/k)^(2*k) * exp(n)). - Vaclav Kotesovec, Jun 17 2026

Links

Table of n, a(n) for n=1..17.

Formula

E.g.f. A(x) satisfies A'(x) = (1 + A(x)*A'(x))^k, with A(0)=0.

Let R(x) = Series_Reversion( (1 - (1-(2*k-1)*x) * (1-x)^(2*k-1))/(2*(2*k-1)) ).

A(x) = R(x) * (1-R(x))^(k-1).

For k > 0 and n > 1, k divides n! * [x^n] A(x).

From Vaclav Kotesovec, Jun 17 2026: (Start)

Recurrence: 72097*a(n) = 28*(249059*n - 1019181)*a(n-1) - 84*(3440899*n^2 - 31579933*n + 73833938)*a(n-2) + 224*(29706005*n^3 - 453218640*n^2 + 2333109877*n - 4050686268)*a(n-3) - 112*(820516025*n^4 - 18321149490*n^3 + 154558838539*n^2 - 583740201282*n + 832645322640)*a(n-4) + 1344*(566484737*n^5 - 17217486965*n^4 + 210173455457*n^3 - 1287947706871*n^2 + 3961887576626*n - 4893807577884)*a(n-5) - 64*(54743609137*n^6 - 2159675281113*n^5 + 35555791047427*n^4 - 312683235834123*n^3 + 1549133056421860*n^2 - 4099499755300740*n + 4527008744162175)*a(n-6) + 52706752*(2*n - 15)*(4*n - 29)*(4*n - 27)*(8*n - 59)*(8*n - 57)*(8*n - 55)*(8*n - 53)*a(n-7).

a(n) ~ 2^(17*n - 17/2) * 7^(n - 3/2) * n^(n-2) / (6561 * exp(n) * 72097^(n - 3/2)). (End)

Prog

(PARI) my(k=4, N=20, x='x+O('x^N), R=serreverse((1-(1-(2*k-1)*x)*(1-x)^(2*k-1))/(2*(2*k-1)))); Vec(serlaplace(R*(1-R)^(k-1)))

Crossrefs

Cf. A001147, A395195, A397131.

Sequence in context: A391550 A377528 A324959 * A098630 A396838 A377629

Adjacent sequences: A397129 A397130 A397131 * A397156 A397157 A397158

Keyword

nonn

Author

Seiichi Manyama, Jun 17 2026

Status

approved