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%I #73 Jun 03 2022 13:00:20
%S 1,1,7,115,3451,164731,11467387,1096832395,138027417451,
%T 22111390122811,4393756903239067,1060590528331645675,
%U 305686632592587314251,103695663062502304228891,40895823706632785802087547,18554695374154504939196298955,9596336362873294022956267703851
%N E.g.f.: Sum_{n>=0} (1 - exp(-n*x))^n.
%C Compare to the trivial identity: exp(x) = Sum_{n>=0} (1 - exp(-x))^n.
%C Compare to the e.g.f. of A092552: Sum_{n>=1} (1 - exp(-n*x))^n/n.
%C From _Arvind Ayyer_, Oct 25 2020: (Start)
%C a(n) is also the number of acyclic orientations with unique sink of the complete bipartite graph K_{n,n+1}
%C 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)
%C 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
%H Vincenzo Librandi, <a href="/A220181/b220181.txt">Table of n, a(n) for n = 0..200</a>
%H A. Ayyer, D. Hathcock and P. Tetali, <a href="https://arxiv.org/abs/2010.11236"> Toppleable Permutations, Excedances and Acyclic Orientations</a>, arXiv:2010.11236 [math.CO], 2020. For the precise definition of a toppleable permutation.
%F O.g.f. Sum_{n>=0} n^n * n! * x^n / Product_{k=1..n} (1 + n*k*x).
%F E.g.f. A(x) = Sum_{n>=0} (1 - exp(-n*x))^n satisfies the identities:
%F (1) A(x) = Sum_{n>=1} exp(-n*x) * (1 - exp(-n*x))^(n-1).
%F (2) A(x) = 1 + (1/2) * Sum_{n>=1} (1 - exp(-n*x))^(n-1).
%F (3) A(x) = Sum_{n>=1} Sum_{k>=0} (-1)^k * C(n+k-1,k) * exp(-k*(n+k-1)*x).
%F E.g.f. at offset 1, B(x) = Sum_{n>=1} a(n-1)*x^n/n!, satisfies:
%F (1) B(x) = Sum_{n>=1} (1 - exp(-n*x))^n / n^2.
%F (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.
%F a(n) = Sum_{k=0..n} (-1)^(n-k) * k^n * k! * Stirling2(n,k).
%F a(n) = Sum_{k=1..n+1} ((k-1)!)^2 * Stirling2(n+1,k)^2 / 2 for n>0 with a(0)=1.
%F a(n) = Sum_{k=0..n} k^n * Sum_{j=0..k} (-1)^(n+k-j) * binomial(k,j) * (k-j)^n.
%F a(n) = A048163(n+1)/2 for n>0.
%F Limit n->infinity (a(n)/n!)^(1/n)/n = 1/(exp(1)*(log(2))^2) = 0.7656928576... - _Vaclav Kotesovec_, Jun 21 2013
%F 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
%e O.g.f.: F(x) = 1 + x + 7*x^2 + 115*x^3 + 3451*x^4 + 164731*x^5 +...
%e 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 ...
%e E.g.f.: A(x) = 1 + x + 7*x^2/2! + 115*x^3/3! + 3451*x^4/4! + 164731*x^5/5! +...
%e where the e.g.f. satisfies the identities:
%e (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 +...
%e (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 +...
%e (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 E.g.f. at offset=1 begins:
%e B(x) = x + x^2/2! + 7*x^3/3! + 115*x^4/4! + 3451*x^5/5! + 164731*x^6/6! +...
%e where
%e 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 +...
%e The series B(x) = Sum_{n>=1} (1 - exp(-n*x))^n / n^2 may be rewritten as:
%e B(x) = Pi^2/6 + log(1-exp(-x)) + Sum_{n>=2} (n-1)*exp(-2*n*x)/(2*n) -
%e 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) -+...
%t Flatten[{1,Table[Sum[(-1)^(n-k)*k^n*k!*StirlingS2[n,k],{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Jun 21 2013 *)
%o (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)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n)=n!*polcoeff(sum(k=0, n, (1-exp(-k*x+x*O(x^n)))^k), n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) /* Formula for this sequence with offset=1: */
%o {a(n)=n!*polcoeff(sum(k=1, n, (1-exp(-k*x+x*O(x^n)))^k/k^2), n)}
%o for(n=1, 21, print1(a(n), ", "))
%o (PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
%o {a(n) = sum(k=0,n,(-1)^(n-k)*k^n*k!*Stirling2(n, k))}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
%o {a(n) = if(n==0,1,sum(k=1,n+1,((k-1)!)^2*Stirling2(n+1,k)^2/2))}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n)=sum(k=0,n, k^n*sum(j=0,k, (-1)^(n+k-j)*binomial(k,j)*(k-j)^n))}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A136126, A092552, A048163, A220179, A122399, A187755, A203798, A320096, A338040.
%K nonn,nice
%O 0,3
%A _Paul D. Hanna_, Dec 06 2012
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