A220179 -id:A220179 - cp's OEIS frontend

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

Paul D. Hanna, Dec 06 2012

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

			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) -+...

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.

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

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

{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), ", "))

{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), ", "))

/* 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), ", "))

{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), ", "))

{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), ", "))

{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), ", "))

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

A092552 Let X_{m,n}(q) be the chromatic polynomial of the complete bipartite graph K_{m,n}. Then a(n) is the negative of the coefficient of the linear term of X_{n,n}(q).

Original entry on oeis.org

0, 1, 3, 31, 675, 25231, 1441923, 116914351, 12764590275, 1805409270031, 321113303226243, 70146437009397871, 18462286083671614275, 5762225835975165678031, 2104263061425865873128963, 888881838896989670838028591, 430058409024841744606172532675

Offset: 0

Michael Lugo (mtlugo(AT)mit.edu), Apr 09 2004

nonn

From Arvind Ayyer, Oct 25 2020: (Start)
Equivalently, a(n) is the number of acyclic orientations with a unique sink in K_{n,n}.
a(n) is also the number of toppleable permutations in S_{2n-1}. A toppleable permutation pi in S_{2n-1} satisfies pi_i <= n-1+i for 1 <= i <= n-1 and pi_i >= i-n+1 for n <= i <= 2n-1. The a(3)=3 toppleable permutations in S_3 are 123, 213 and 132. (End)

			a(2) = 3 since the chromatic polynomial of K_{2,2}(q) is q^4-4*q^3+6*q^2-3*q.
E.g.f.: A(x) = x + 3*x^2/2! + 31*x^3/3! + 675*x^4/4! + 25231*x^5/5! +...
where A(x) = (1-exp(-x)) + (1-exp(-2*x))^2/2 + (1-exp(-3*x))^3/3 +... - _Paul D. Hanna_, Dec 06 2012
O.g.f.: F(x) = x + 3*x^2 + 31*x^3 + 675*x^4 + 25231*x^5 +...
where F(x) = x/(1+x) + 2^1*2!*x^2/((1+2*1*x)*(1+2*2*x)) + 3^2*3!*x^3/((1+3*1*x)*(1+3*2*x)*(1+3*3*x)) + 4^3*4!*x^4/((1+4*1*x)*(1+4*2*x)*(1+4*3*x)*(1+4*4*x)) +... - _Paul D. Hanna_, Jan 05 2013

Alois P. Heinz, Table of n, a(n) for n = 0..238
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.

Cf. A008277, A212084, A220179, A136126, A306209.

a:= n-> -coeff(add(Stirling2(n, k) *mul(q-i, i=0..k-1)
             *(q-k)^n, k=1..n), q, 1):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 30 2012

Table[Sum[k!*(k-1)!*StirlingS2[n,k]^2,{k,1,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 21 2013 *)

{a(n)=n!*polcoeff(sum(k=1,n,(1-exp(-k*x+x*O(x^n)))^k/k),n)} \\ Paul D. Hanna, Dec 06 2012
for(n=0,20,print1(a(n),", "))

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=if(n<=0, 0, sum(k=1, n, k!*(k-1)! * Stirling2(n, k)^2))} \\ Paul D. Hanna, Dec 30 2012
for(n=0, 20, print1(a(n), ", "))

{a(n)=if(n<1,0,polcoeff(sum(m=1, n, m^(m-1)*m!*x^m/prod(k=1, m, 1+m*k*x+x*O(x^n))), n))} \\ Paul D. Hanna, Jan 05 2013

From Alois P. Heinz, Apr 30 2012: (Start)
a(n) = (-1) * [q] Sum_{j=1..n} (q-j)^n*S2(n,j)*Product_{i=0..j-1} (q-i).
a(n) = (-1) * A212084(n,2n-1). (End)
E.g.f.: Sum_{n>=1} (1 - exp(-n*x))^n / n. - Paul D. Hanna, Dec 06 2012
a(n) = Sum_{k=1..n} k!*(k-1)! * Stirling2(n, k)^2. - Paul D. Hanna, Dec 30 2012, corrected by Vaclav Kotesovec, Jun 21 2013
O.g.f.: Sum_{n>=1} n^(n-1) * n! * x^n / Product_{k=1..n} (1 + n*k*x). - Paul D. Hanna, Jan 05 2013
a(n) = A136126(2*n-1,n), where triangle A136126(n,k) is the number of permutations of {1,2,...,k+n} having excedance set {1,2,...,k}. - Paul D. Hanna, Feb 01 2013
a(n) ~ sqrt(Pi) * n^(2*n-1/2) / (sqrt(1-log(2)) * exp(2*n) * (log(2))^(2*n)). - Vaclav Kotesovec, Nov 07 2014
a(n) = A306209(2n-1,n-1) for n > 0. - Alois P. Heinz, Feb 01 2019

A242228 a(n) = Sum_{k=1..n} k^(2n-1) k! * Stirling2(n,k).

Original entry on oeis.org

1, 17, 1651, 473741, 300257371, 355743405917, 706872713310331, 2182548723605418941, 9894910566488309801851, 63052832687428562206049117, 545439670961897317869306191611, 6226501736967631584015448186252541, 91619831483112536750163352484302283131

Offset: 1

Vaclav Kotesovec, May 08 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..160

Cf. A000629, A220179, A229260, A244585.

Table[Sum[k^(2*n-1) * k! * StirlingS2[n,k], {k,1,n}], {n,1,20}]

a(n) ~ c * d^n * (n!)^3 / n^2, where d = r^3*(1+exp(2/r)) = 7.8512435106631367719817991716164612615296980032514..., r = 0.94520217245242431308104743874492469552738... is the root of the equation (1+exp(-2/r))*LambertW(-exp(-1/r)/r) = -1/r, and c = 0.15095210978787998524366903417512193343948127919...

E.g.f.: Sum_{k>=1} (exp(k^2*x) - 1)^k / k. - Seiichi Manyama, Jun 19 2024

cp's OEIS Frontend

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

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A092552 Let X_{m,n}(q) be the chromatic polynomial of the complete bipartite graph K_{m,n}. Then a(n) is the negative of the coefficient of the linear term of X_{n,n}(q).

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A242228 a(n) = Sum_{k=1..n} k^(2*n-1) * k! * Stirling2(n,k).

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A373856 a(n) = Sum_{k=1..n} k! * k^(2*n-1) * |Stirling1(n,k)|.

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A373858 a(n) = Sum_{k=1..n} k! * k^(2*n-1) * Stirling1(n,k).

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A242228 a(n) = Sum_{k=1..n} k^(2n-1) k! * Stirling2(n,k).

A373856 a(n) = Sum_{k=1..n} k! * k^(2n-1) |Stirling1(n,k)|.

A373858 a(n) = Sum_{k=1..n} k! * k^(2n-1) Stirling1(n,k).