A136729 E.g.f.: A(x) = [ exp(x)/(5 - 4*exp(x)) ]^(1/5).

1, 1, 5, 49, 701, 13177, 306821, 8520289, 274808525, 10095533833, 416131518293, 19017974164465, 954399901374749, 52173428322993433, 3085965087129209381, 196360349627069553793, 13374490368820471936109, 970904530181260115741737

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

E.g.f. A(x) satisfies: A(x) = 1 + integral( A(x)^6 * exp(-x) ).

O.g.f.: 1/(1 - x/(1-4*x/(1 - 6*x/(1-8*x/(1 - 11*x/(1-16*x/(1 - 16*x/(1-24*x/(1 - 21*x/(1-32*x/(1 - ...)))))))))), a continued fraction.

G.f.: 1/G(0) where G(k) = 1 - x*(5*k+1)/( 1 - 4*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 23 2013

a(n) ~ n! * sqrt(5-sqrt(5))*Gamma(4/5) / (2*Log[5/4]^(n+1/5) * 2^(9/10)*n^(4/5)*Pi). - Vaclav Kotesovec, Sep 22 2013

a(n) = 1 + 4 * Sum_{k=1..n-1} (binomial(n,k) - 1) * a(k). - Ilya Gutkovskiy, Jul 09 2020

From Seiichi Manyama, Nov 15 2023: (Start)

a(n) = Sum_{k=0..n} (-1)^(n-k) * (Product_{j=0..k-1} (5*j+1)) * Stirling2(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * (4*k/n - 5) * binomial(n,k) * a(n-k).

a(0) = 1; a(n) = a(n-1) + 4*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)

Mathematica

CoefficientList[Series[(E^x/(5-4*E^x))^(1/5), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 22 2013 *)

Prog

(PARI) a(n)=n!*polcoeff((exp(x +x*O(x^n))/(5-4*exp(x +x*O(x^n))))^(1/5), n)
(PARI) /* As solution to integral equation: */ a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+intformal(A^6*exp(-x+x*O(x^n)))); n!*polcoeff(A, n)

Crossrefs

Variants: A014307, A136727, A136728.

Sequence in context: A089914 A267220 A052142 * A102773 A028575 A368438

Adjacent sequences: A136726 A136727 A136728 * A136730 A136731 A136732

Keyword

nonn

Author

Paul D. Hanna, Jan 24 2008

Status

approved