A235371 E.g.f. satisfies: A'(x) = A(x)^6 * A(-x) with A(0) = 1.

1, 1, 5, 53, 745, 14761, 342125, 9919133, 320525905, 12231596881, 507077877845, 23997083985413, 1214302628958265, 68589494705086201, 4098200887304658365, 269019686917890755693, 18535055580911326897825, 1386438585493229417193121, 108209001114760469546923685

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..358

Formula

E.g.f.: 1/(1 - 4*Series_Reversion( Integral (1 - 16*x^2)^(1/4) dx ))^(1/4).

Lim_{n->infinity} (a(n)/n!)^(1/n) = 12*sqrt(2) * GAMMA(3/4)^2 / Pi^(3/2) = 4.57655858101... - Vaclav Kotesovec, Jan 28 2014

Example

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 53*x^3/3! + 745*x^4/4! + 14761*x^5/5! + ...
Related series.
A(x)^6 = 1 + 6*x + 60*x^2/2! + 888*x^3/3! + 17040*x^4/4! + 407136*x^5/5! + ...
Note that 1 - 1/A(x)^4 is an odd function:
1 - 1/A(x)^4 = 4*x + 32*x^3/3! + 4864*x^5/5! + 2250752*x^7/7! + ...
where Series_Reversion((1 - 1/A(x)^4)/4) = Integral (1-16*x^2)^(1/4) dx.

Mathematica

CoefficientList[1/(1 - 4*InverseSeries[Series[Integrate[(1-16*x^2)^(1/4), x], {x, 0, 20}], x])^(1/4), x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 28 2014 *)

Prog

(PARI) {a(n)=local(A=1); for(i=0, n, A=1+intformal(A^6*subst(A, x, -x)^1 +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); A=1/(1-4*serreverse(intformal((1-16*x^2 +x*O(x^n))^(1/4))))^(1/4); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A235370, A235372, A235373, A235374.

Sequence in context: A123788 A333096 A036910 * A036916 A118583 A090360

Adjacent sequences: A235368 A235369 A235370 * A235372 A235373 A235374

Keyword

nonn

Author

Paul D. Hanna, Jan 07 2014

Status

approved