A136728 E.g.f.: A(x) = (exp(x)/(4 - 3*exp(x)))^(1/4).

1, 1, 4, 31, 349, 5146, 93799, 2036161, 51283894, 1470035101, 47250248569, 1683031711516, 65800765032589, 2801364476781781, 129003301751229364, 6389120632590635971, 338644807090096148809, 19126604338708282552186

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..373

Formula

E.g.f. A(x) satisfies: A(x) = 1 + integral( A(x)^5 * exp(-x) ).

O.g.f.: 1/(1 - x/(1-3*x/(1 - 5*x/(1-6*x/(1 - 9*x/(1-9*x/(1 - 13*x/(1-12*x/(1 - 17*x/(1-15*x/(1 - ...)))))))))), a continued fraction.

G.f.: 1/G(0) where G(k) = 1 - x*(4*k+1)/( 1 - 3*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 23 2013

a(n) ~ n! * Gamma(3/4)/(sqrt(2)*3^(1/4)*n^(3/4)*Pi*log(4/3)^(n+1/4)). - Vaclav Kotesovec, Jun 15 2013

a(n) = 1 + 3 * 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} (4*j+1)) * Stirling2(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * (3*k/n - 4) * binomial(n,k) * a(n-k).

a(0) = 1; a(n) = a(n-1) + 3*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)

Mathematica

CoefficientList[Series[(E^x/(4-3*E^x))^(1/4), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 15 2013 *)

Prog

(PARI) a(n)=n!*polcoeff((exp(x +x*O(x^n))/(4-3*exp(x +x*O(x^n))))^(1/4), 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^4*exp(-x+x*O(x^n)))); n!*polcoeff(A, n)

Crossrefs

Cf. A201354, variants: A014307, A136727, A136729.

Sequence in context: A107725 A145160 A129271 * A321031 A102757 A295254

Adjacent sequences: A136725 A136726 A136727 * A136729 A136730 A136731

Keyword

nonn

Author

Paul D. Hanna, Jan 24 2008

Status

approved