OFFSET
1,2
COMMENTS
In general, for k >= 1, if e.g.f. A(x) satisfies A'(x) = 1/(1 - A(x)*A'(x))^k, then a(n) ~ k^(1/2) * (1 + 1/k)^((2*n-1)*k) * ((4*k+2)/(1 + (1 + 1/k)^(2*k)))^(n - 3/2) * n^(n-2) / exp(n). - Vaclav Kotesovec, Jun 17 2026
FORMULA
E.g.f. A(x) satisfies A'(x) = 1/(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))/(2*(2*k+1)) ).
A(x) = R(x) * (1-R(x))^k.
For k > 0 and n > 1, k divides n! * [x^n] A(x).
From Vaclav Kotesovec, Jun 17 2026: (Start)
Recurrence: 456161*a(n) = 18*(3583752*n - 16714117)*a(n-1) - 12*(332471412*n^2 - 3430392741*n + 9010347233)*a(n-2) + 72*(1957669560*n^3 - 33204879585*n^2 + 190103590395*n - 367191374338)*a(n-3) - 48*(64817234370*n^4 - 1594113649425*n^3 + 14826659769540*n^2 - 61794527876739*n + 97353224190766)*a(n-4) + 288*(152551490616*n^5 - 5066973793215*n^4 + 67681148488830*n^3 - 454403862326493*n^2 + 1533296126513406*n - 2079988483629760)*a(n-5) - 64*(6056435559132*n^6 - 259356683749437*n^5 + 4641685735805235*n^4 - 44437290254231337*n^3 + 240004805156502993*n^2 - 693345916284782706*n + 836963493377960135)*a(n-6) + 576*(3391163284752*n^7 - 181151361522486*n^6 + 4152242778865038*n^5 - 52938706059665226*n^4 + 405445982942328570*n^3 - 1865334259707568128*n^2 + 4773253351031447750*n - 5240771075336188425)*a(n-7) - 900000000*(3*n - 25)*(3*n - 23)*(9*n - 77)*(9*n - 76)*(9*n - 74)*(9*n - 73)*(9*n - 71)*(9*n - 70)*a(n-8).
a(n) ~ 2^(n - 33/2) * 3^(2*n-3) * 5^(8*n-4) * n^(n-2) / (exp(n) * 456161^(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))/(2*(2*k+1)))); Vec(serlaplace(R*(1-R)^k))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jun 17 2026
STATUS
approved