OFFSET
1,2
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: 4825*a(n) = 7*(56442*n - 208655)*a(n-1) - 7*(1924440*n^2 - 16132256*n + 34406621)*a(n-2) + 7*(34970320*n^3 - 491584856*n^2 + 2327414474*n - 3709967179)*a(n-3) - 14*(178593240*n^4 - 3700262048*n^3 + 28906339014*n^2 - 100898558488*n + 132759748449)*a(n-4) + 28*(486058440*n^5 - 13787992204*n^4 + 156760547062*n^3 - 892885832993*n^2 + 2547801577622*n - 2913532853319)*a(n-5) - 262144*(7*n - 46)*(7*n - 45)*(7*n - 44)*(7*n - 43)*(7*n - 41)*(7*n - 40)*a(n-6).
a(n) ~ 2^(13*n - 15/2) * 7^(n - 3/2) * n^(n-2) / (3^(11/2) * 5^(2*n-3) * exp(n) * 193^(n - 3/2)). (End)
PROG
(PARI) my(k=3, 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