login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

E.g.f.: A(x) = [ exp(x)/(5 - 4*exp(x)) ]^(1/5).
7

%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