A258994 E.g.f.: A'(x) = 1 + A(x)^6, with A(0)=1.

1, 2, 12, 192, 4272, 124992, 4531392, 195869952, 9832326912, 562125837312, 36056880110592, 2564230500421632, 200237330428342272, 17032391106795159552, 1567547894591436275712, 155196096043697480466432, 16447362605632117421309952, 1857733260790463501532659712

Offset: 0

Author

Vaclav Kotesovec, Jun 16 2015

Keywords

• nonn

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).

Examples

			A(x) = 1 + 2*x + 12*x^2/2! + 192*x^3/3! + 4272*x^4/4! + 124992*x^5/5! + ...
A'(x) = 2 + 12*x + 96*x^2 + 712*x^3 + 5208*x^4 + 188808*x^5/5 + ...
1 + A(x)^6 = 2 + 12*x + 96*x^2 + 712*x^3 + 5208*x^4 + 188808*x^5/5 + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..50

Crossrefs

Cf. A000831 (k=2), A258969 (k=3), A258970 (k=4), A258971 (k=5), A258927.

Programs

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^6-D[egf,x]],x],nmax]==ConstantArray[0,nmax]][[1]]

PARI

{a(n) = local(A=1); A = 1 + serreverse( intformal( 1/(1 + (1+x)^6 +x*O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Jun 16 2015

Formula

a(n) ~ n! * d^(n+1/5) / (5^(1/5) * Gamma(1/5) * n^(4/5)), where d = 1 / Sum_{j>=1} (-1)^(j+1)/(6*j-1) = 6/(Pi - sqrt(3)*log(2+sqrt(3))) = 6.97224737278326506475991855023425659249063565...

E.g.f.: 1 + Series_Reversion( Integral 1/(1 + (1+x)^6) dx ). - Paul D. Hanna, Jun 16 2015