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
Examples
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) -+...
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.
Programs
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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 *) -
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), ", "))
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
Comments