A136729 - OEIS

%I #25 Nov 15 2023 08:04:01

%S 1,1,5,49,701,13177,306821,8520289,274808525,10095533833,416131518293,

%T 19017974164465,954399901374749,52173428322993433,3085965087129209381,

%U 196360349627069553793,13374490368820471936109,970904530181260115741737

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

%H Vincenzo Librandi, <a href="/A136729/b136729.txt">Table of n, a(n) for n = 0..200</a>

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

%F 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.

%F 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

%F 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

%F a(n) = 1 + 4 * Sum_{k=1..n-1} (binomial(n,k) - 1) * a(k). - _Ilya Gutkovskiy_, Jul 09 2020

%F From _Seiichi Manyama_, Nov 15 2023: (Start)

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

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

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

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

%o (PARI) a(n)=n!*polcoeff((exp(x +x*O(x^n))/(5-4*exp(x +x*O(x^n))))^(1/5),n)

%o (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)

%Y Variants: A014307, A136727, A136728.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 24 2008