A258971 E.g.f.: A'(x) = 1 + A(x)^5, with A(0)=1.

1, 2, 10, 130, 2330, 54770, 1591690, 55065250, 2209888250, 100922263250, 5167670934250, 293215490277250, 18260340583516250, 1238269550334211250, 90824251513716786250, 7164531681653318001250, 604824006980892825496250, 54406894886223009690031250

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

Cf. A000831 (k=2), A258969 (k=3), A258970 (k=4), A258994 (k=6), A258925.

Sequence in context: A119191 A125993 A185952 * A011805 A294350 A194158

Adjacent sequences: A258968 A258969 A258970 * A258972 A258973 A258974

Keyword

nonn

Author

Vaclav Kotesovec, Jun 15 2015

Status

approved