OFFSET
0,2
COMMENTS
In general, for k>1, if e.g.f. satisfies A'(x) = 1 + A(x)^k, with A(0)=1, then a(n) ~ n! * d^(n + 1/(k-1)) / ((k-1)^(1/(k-1)) * Gamma(1/(k-1)) * n^(1-1/(k-1))), where d = 1 / Sum_{j>=1} (-1)^(j+1)/(k*j-1).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..80
FORMULA
a(n) ~ n! * d^(n+1/4) / (4^(1/4) * Gamma(1/4) * n^(3/4)), where d = 1 / Sum_{j>=1} (-1)^(j+1)/(5*j-1) = 40*sqrt(5-sqrt(5)) / (8*sqrt(2)*Pi + sqrt(5+sqrt(5)) * ((9-5*sqrt(5))*log(2) + (sqrt(5)-5)*log(7+3*sqrt(5)))) = 5.53569595526739362969262739469167643400611216649309306882558956...
E.g.f.: 1 + Series_Reversion( Integral 1/(1 + (1+x)^5) dx ). - Paul D. Hanna, Jun 16 2015
EXAMPLE
A(x) = 1 + 2*x + 10*x^2/2! + 130*x^3/3! + 2330*x^4/4! + 54770*x^5/5! + ...
A'(x) = 2 + 10*x + 65*x^2 + 1165*x^3/3 + 27385*x^4/12 + 159169*x^5/12 + ...
1 + A(x)^5 = 2 + 10*x + 65*x^2 + 1165*x^3/3 + 27385*x^4/12 + 159169*x^5/12 + ...
MATHEMATICA
nmax=20; Subscript[a, 0]=1; egf=Sum[Subscript[a, k]*x^k, {k, 0, nmax+1}]; Table[Subscript[a, k]*k!, {k, 0, nmax}] /.Solve[Take[CoefficientList[Expand[1+egf^5-D[egf, x]], x], nmax]==ConstantArray[0, nmax]][[1]]
PROG
(PARI) {a(n) = local(A=1); A = 1 + serreverse( intformal( 1/(1 + (1+x)^5 +x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Jun 16 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Jun 15 2015
STATUS
approved