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A201368 E.g.f.: 4*exp(4*x) / (5 - exp(4*x)). 4
1, 5, 30, 230, 2280, 28280, 421680, 7336880, 145879680, 3263031680, 81097294080, 2217097729280, 66122900014080, 2136392343342080, 74335250629908480, 2771225281718343680, 110198981079416340480, 4655992415884353044480, 208291013498682750074880, 9835804726301090178990080 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..19.

FORMULA

O.g.f.: A(x) = Sum_{n>=0} n! * 5^n*x^n / Product_{k=0..n} (1+4*k*x).

O.g.f.: A(x) = 1/(1 - 5*x/(1-x/(1 - 10*x/(1-2*x/(1 - 15*x/(1-3*x/(1 - 20*x/(1-4*x/(1 - 25*x/(1-5*x/(1 - ...))))))))))), a continued fraction.

a(n) = Sum_{k=0..n} (-4)^(n-k) * 5^k * Stirling2(n,k) * k!.

a(n) = Sum_{k, 0<=k<=n} A123125(n,k)*5^k. - Philippe Deléham, Nov 30 2011

a(n) ~ n! * (4/log(5))^(n+1). - Vaclav Kotesovec, Jun 13 2013

a(n) = 4^n*log(5) * int {x = 0..inf} (ceiling(x))^n * 5^(-x) dx. - Peter Bala, Feb 06 2015

EXAMPLE

E.g.f.: E(x) = 1 + 5*x + 30*x^2/2! + 230*x^3/3! + 2280*x^4/4! + 28280*x^5/5! +...

O.g.f.: A(x) = 1 + 5*x + 30*x^2 + 230*x^3 + 2280*x^4 + 28280*x^5 +...

where A(x) = 1 + 5*x/(1+4*x) + 2!*5^2*x^2/((1+4*x)*(1+8*x)) + 3!*5^3*x^3/((1+4*x)*(1+8*x)*(1+12*x)) + 4!*5^4*x^4/((1+4*x)*(1+8*x)*(1+12*x)*(1+16*x)) +...

MATHEMATICA

Table[Sum[(-4)^(n-k)*5^k*StirlingS2[n, k]*k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 13 2013 *)

PROG

(PARI) {a(n)=n!*polcoeff(4*exp(4*x+x*O(x^n))/(5 - exp(4*x+x*O(x^n))), n)}

(PARI) {a(n)=polcoeff(sum(m=0, n, 5^m*m!*x^m/prod(k=1, m, 1+4*k*x+x*O(x^n))), n)}

(PARI) {Stirling2(n, k)=if(k<0|k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}

{a(n)=sum(k=0, n, (-4)^(n-k)*5^k*Stirling2(n, k)*k!)}

(Sage)

@CachedFunction

def BB(n, k, x):  # modified cardinal B-splines

    if n == 1: return 0 if (x < 0) or (x >= k) else 1

    return x*BB(n-1, k, x) + (n*k-x)*BB(n-1, k, x-k)

def EulerianPolynomial(n, k, x):

    if n == 0: return 1

    return add(BB(n+1, k, k*m+1)*x^m for m in (0..n))

[5^n*EulerianPolynomial(n, 1, 1/5) for n in (0..19)]   # Peter Luschny, May 04 2013

(PARI) x='x+O('x^66); Vec(serlaplace(4*exp(4*x)/(5-exp(4*x)))) \\ Joerg Arndt, May 06 2013

CROSSREFS

Cf. A201365, A201366, A201367.

Sequence in context: A318920 A167892 A144498 * A072213 A257741 A279155

Adjacent sequences:  A201365 A201366 A201367 * A201369 A201370 A201371

KEYWORD

nonn,easy

AUTHOR

Paul D. Hanna, Nov 30 2011

STATUS

approved

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Last modified October 16 09:27 EDT 2018. Contains 316262 sequences. (Running on oeis4.)