A220181 E.g.f.: Sum_{n>=0} (1 - exp(-n*x))^n.

1, 1, 7, 115, 3451, 164731, 11467387, 1096832395, 138027417451, 22111390122811, 4393756903239067, 1060590528331645675, 305686632592587314251, 103695663062502304228891, 40895823706632785802087547, 18554695374154504939196298955, 9596336362873294022956267703851

Offset

0,3

Comments

Compare to the trivial identity: exp(x) = Sum_{n>=0} (1 - exp(-x))^n.

Compare to the e.g.f. of A092552: Sum_{n>=1} (1 - exp(-n*x))^n/n.

From Arvind Ayyer, Oct 25 2020: (Start)

a(n) is also the number of acyclic orientations with unique sink of the complete bipartite graph K_{n,n+1}

a(n) is also the number of toppleable permutations in S_{2n}. A toppleable permutation pi in S_{2n} satisfies pi_i <= n-1+i for 1 <= i <= n+1 and pi_i >= i-n for n+2 <= i <= 2n. (End)

Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 0, 3, 0, 0, 1, 1, 0, 3, 0, 0, 1, 1, 0, 3, 0, 0, 1 ...], with an apparent period of 6. Cf. A122399. - Peter Bala, Jun 01 2022

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

A. Ayyer, D. Hathcock and P. Tetali, Toppleable Permutations, Excedances and Acyclic Orientations, arXiv:2010.11236 [math.CO], 2020. For the precise definition of a toppleable permutation.

Formula

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

E.g.f. A(x) = Sum_{n>=0} (1 - exp(-n*x))^n satisfies the identities:

(1) A(x) = Sum_{n>=1} exp(-n*x) * (1 - exp(-n*x))^(n-1).

(2) A(x) = 1 + (1/2) * Sum_{n>=1} (1 - exp(-n*x))^(n-1).

(3) A(x) = Sum_{n>=1} Sum_{k>=0} (-1)^k * C(n+k-1,k) * exp(-k*(n+k-1)*x).

E.g.f. at offset 1, B(x) = Sum_{n>=1} a(n-1)*x^n/n!, satisfies:

(1) B(x) = Sum_{n>=1} (1 - exp(-n*x))^n / n^2.

(2) B(x) = Pi^2/6 + log(1-exp(-x)) + Sum_{k>=2} Sum_{n>=k} (-1)^k * C(n-1,k-1) * exp(-k*n*x)/(k*n), a convergent series for x>0.

a(n) = Sum_{k=0..n} (-1)^(n-k) * k^n * k! * Stirling2(n,k).

a(n) = Sum_{k=1..n+1} ((k-1)!)^2 * Stirling2(n+1,k)^2 / 2 for n>0 with a(0)=1.

a(n) = Sum_{k=0..n} k^n * Sum_{j=0..k} (-1)^(n+k-j) * binomial(k,j) * (k-j)^n.

a(n) = A048163(n+1)/2 for n>0.

Limit n->infinity (a(n)/n!)^(1/n)/n = 1/(exp(1)*(log(2))^2) = 0.7656928576... - Vaclav Kotesovec, Jun 21 2013

a(n) ~ sqrt(Pi) * n^(2*n+1/2) / (sqrt(1-log(2)) * exp(2*n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, May 13 2014

Example

O.g.f.: F(x) = 1 + x + 7*x^2 + 115*x^3 + 3451*x^4 + 164731*x^5 +...
where F(x) = 1 + x/(1+x) + 2^2*2!*x^2/((1+2*1*x)*(1+2*2*x)) + 3^3*3!*x^3/((1+3*1*x)*(1+3*2*x)*(1+3*3*x)) + 4^4*4!*x^4/((1+4*1*x)*(1+4*2*x)*(1+4*3*x)*(1+4*4*x)) +...
...
E.g.f.: A(x) = 1 + x + 7*x^2/2! + 115*x^3/3! + 3451*x^4/4! + 164731*x^5/5! +...
where the e.g.f. satisfies the identities:
(1) A(x) = 1 + (1-exp(-x)) + (1-exp(-2*x))^2 + (1-exp(-3*x))^3 + (1-exp(-4*x))^4 + (1-exp(-5*x))^5 + (1-exp(-6*x))^6 +...
(2) A(x) = exp(-x) + exp(-2*x)*(1-exp(-2*x)) + exp(-3*x)*(1-exp(-3*x))^2 + exp(-4*x)*(1-exp(-4*x))^3 + exp(-5*x)*(1-exp(-5*x))^4 + exp(-6*x)*(1-exp(-6*x))^5 +...
(3) 2*A(x) = 2 + (1-exp(-2*x)) + (1-exp(-3*x))^2 + (1-exp(-4*x))^3 + (1-exp(-5*x))^4 + (1-exp(-6*x))^5 + (1-exp(-7*x))^6 +...
E.g.f. at offset=1 begins:
B(x) = x + x^2/2! + 7*x^3/3! + 115*x^4/4! + 3451*x^5/5! + 164731*x^6/6! +...
where
B(x) = (1-exp(-x)) + (1-exp(-2*x))^2/2^2 + (1-exp(-3*x))^3/3^2 + (1-exp(-4*x))^4/4^2 + (1-exp(-5*x))^5/5^2 + (1-exp(-6*x))^6/6^2 +...
The series  B(x) = Sum_{n>=1} (1 - exp(-n*x))^n / n^2  may be rewritten as:
B(x) = Pi^2/6 + log(1-exp(-x)) + Sum_{n>=2} (n-1)*exp(-2*n*x)/(2*n) -
Sum_{n>=3} C(n-1,2)*exp(-3*n*x)/(3*n) + Sum_{n>=4} C(n-1,3)*exp(-4*n*x)/(4*n) -+...

Mathematica

Flatten[{1, Table[Sum[(-1)^(n-k)*k^n*k!*StirlingS2[n, k], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jun 21 2013 *)

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, m^m*m!*x^m/prod(k=1, m, 1+m*k*x+x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=n!*polcoeff(sum(k=0, n, (1-exp(-k*x+x*O(x^n)))^k), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Formula for this sequence with offset=1: */
{a(n)=n!*polcoeff(sum(k=1, n, (1-exp(-k*x+x*O(x^n)))^k/k^2), n)}
for(n=1, 21, print1(a(n), ", "))
(PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = sum(k=0, n, (-1)^(n-k)*k^n*k!*Stirling2(n, k))}
for(n=0, 20, print1(a(n), ", "))
(PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0, 1, sum(k=1, n+1, ((k-1)!)^2*Stirling2(n+1, k)^2/2))}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, n, k^n*sum(j=0, k, (-1)^(n+k-j)*binomial(k, j)*(k-j)^n))}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A136126, A092552, A048163, A220179, A122399, A187755, A203798, A320096, A338040.

Sequence in context: A082487 A081798 A063399 * A328813 A178297 A285394

Adjacent sequences: A220178 A220179 A220180 * A220182 A220183 A220184

Keyword

nonn,nice

Author

Paul D. Hanna, Dec 06 2012

Status

approved