A014479 Exponential generating function = (1+2*x)/(1-2*x)^3.

1, 8, 72, 768, 9600, 138240, 2257920, 41287680, 836075520, 18579456000, 449622835200, 11771943321600, 331576403558400, 9998303861145600, 321374052679680000, 10969567664799744000, 396275631890890752000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..395
• Ben Adenbaum, Jennifer Elder, Pamela E. Harris, and J. Carlos Martínez Mori, Boolean intervals in the weak Bruhat order of a finite Coxeter group, arXiv:2403.07989 [math.CO], 2024. See p. 8.

Crossrefs

Cf. A001620, A014477, A187735.

Programs

Maple

seq(add(count(Composition(k))*count(Permutation(k)),k=1..n),n=1..17); # Zerinvary Lajos, Oct 17 2006
seq(2^n*(n+1)^2*n!, n=0..30); # Robert Israel, Oct 28 2015

Mathematica

Table[2^n (n+1)^2 n!, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 28 2015 *)

PARI

{a(n)=polcoeff( sum(m=0,n,(2*m+1)^(m+1)*x^m / (1 + (2*m+1)*x +x*O(x^n))^(m+1)),n)} \\ Paul D. Hanna, Jan 02 2013
for(n=0,20,print1(a(n),", "))

PARI

vector(30, n, n--; n!*(n+1)^2*2^n) \\ Altug Alkan, Oct 28 2015

Formula

a(n) = A014477(n) * n!. - Franklin T. Adams-Watters, Nov 02 2006

G.f.: Sum_{n>=0} (2*n+1)^(n+1) * x^n / (1 + (2*n+1)*x)^(n+1). - Paul D. Hanna, Jan 02 2013

From Vladimir Reshetnikov, Oct 28 2015: (Start)

a(n) = 2^n*(n+1)^2*n!.

Recurrence: a(0) = 1, n*a(n) = 2*(n+1)^2*a(n-1). (End)

From Amiram Eldar, Dec 10 2022: (Start)

Sum_{n>=0} 1/a(n) = 2*(Ei(1/2) - gamma + log(2)), where Ei(x) is the exponential integral and gamma is Euler's constant (A001620).

Sum_{n>=0} (-1)^n/a(n) = 2*(gamma - Ei(-1/2) - gamma - log(2)). (End)