A259113 E.g.f. satisfies: A(x) = Integral 1 + A(x)^8 dx.

1, 40320, 18598035456000, 474009962689446543360000, 170149872975531014630262649651200000, 442695618409212548301531680485487369256960000000, 5620045472937667963281036681944526735620775198955929600000000

Offset: 0

Author

Vaclav Kotesovec, Jun 18 2015

Keywords

• nonn

Comments

In general, for k>2, if e.g.f. satisfies A(x) = Integral 1 + A(x)^k dx, then a(n) ~ k^(k/(k-1)) * n^(1/(k-1)) * (k*n)! * (k*sin(Pi/k)/Pi)^(k*n + k/(k-1)) / ((k-1)^(1/(k-1)) * Gamma(1/(k-1))).

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..56

Crossrefs

Cf. A258880 (k=3), A258901 (k=4), A258925 (k=5), A258927 (k=6), A259112 (k=7).

Programs

PARI

{a(n) = local(A=x); A = serreverse( intformal( 1/(1 + x^8 + O(x^(8*n+2))) ) ); (8*n+1)!*polcoeff(A, 8*n+1)}
for(n=0, 20, print1(a(n), ", ")) \\ after Paul D. Hanna

Formula

a(n) ~ 2^(24*n+48/7) * n^(1/7) * (sin(Pi/8)/Pi)^(8*n+8/7) * (8*n)! / (7^(1/7) * GAMMA(1/7)).

a(n) ~ 2^(16*n+40/7) * (2-sqrt(2))^(4*n+4/7) * n^(1/7) * (8*n)! / (7^(1/7) * GAMMA(1/7) * Pi^(8*n+8/7)).