A203798 -id:A203798 - 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

A187755 E.g.f.: Sum_{n>=0} (1 - exp(-n^2*x))^n.

Original entry on oeis.org

1, 1, 31, 3991, 1340251, 929043391, 1153715889691, 2333670966674671, 7180487882511523051, 31919495229412870788031, 196909477461357591810377851, 1632140626754602443266222263951, 17701927686793740884448652685728651, 245721757633690118910277310669218472671

Offset: 0

Paul D. Hanna, Jan 03 2013

nonn

			O.g.f.: F(x) = 1 + x + 31*x^2 + 3991*x^3 + 1340251*x^4 + 929043391*x^5 +...
where
F(x) = 1 + x/(1+x) + 2^4*2!*x^2/((1+2^2*1*x)*(1+2^2*2*x)) + 3^6*3!*x^3/((1+3^2*1*x)*(1+3^2*2*x)*(1+3^2*3*x)) + 4^8*4!*x^4/((1+4^2*1*x)*(1+4^2*2*x)*(1+4^2*3*x)*(1+4^2*4*x)) +...
...
E.g.f.: A(x) = 1 + x + 31*x^2/2! + 3991*x^3/3! + 1340251*x^4/4! +...
where
A(x) = 1 + (1-exp(-x)) + (1-exp(-2^2*x))^2 + (1-exp(-3^2*x))^3 + (1-exp(-4^2*x))^4 + (1-exp(-5^2*x))^5 + (1-exp(-6^2*x))^6 +...

Seiichi Manyama, Table of n, a(n) for n = 0..162

Cf. A220181, A203798, A229260.

Flatten[{1,Table[Sum[(-1)^(n-k) * k^(2*n) * k! * StirlingS2[n,k], {k,0,n}], {n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)

{a(n)=n!*polcoeff(sum(k=0, n, (1-exp(-k^2*x+x*O(x^n)))^k), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=polcoeff(sum(m=0, n, m^(2*m)*m!*x^m/prod(k=1, m, 1+m^2*k*x+x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = sum(k=0, n, (-1)^(n-k)*k^(2*n)*k!*stirling(n, k, 2))}
for(n=0, 20, print1(a(n), ", "))

O.g.f.: Sum_{n>=0} n^(2*n) * n! * x^n / Product_{k=1..n} (1 + n^2*k*x).
a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(2*n) * k! * Stirling2(n,k).
a(n) == 1 (mod 10) for n>=0.
a(n) == 31 (mod 60) for n>=2.
a(n) ~ c * d^n * (n!)^3 / n, where d = 6.8312860494079582446988970296645779575650627187418208311407895492635... and c = 0.192038502554748256318271067254582378566365276592... . - Vaclav Kotesovec, May 08 2014

A242229 a(n) = Sum_{k=0..n} k^(3n) k! * StirlingS2(n,k).

Original entry on oeis.org

1, 1, 129, 121171, 421842405, 3921960731851, 80097035486409669, 3154805675402432477371, 218356776433458097793841045, 24765902586563160053438320367371, 4359137561016969073655241431827801509, 1139916274502953599866121961715757905518171

Offset: 0

Vaclav Kotesovec, May 08 2014

Seiichi Manyama, Table of n, a(n) for n = 0..125

Cf. A203798, A242228

Cf. A000670, A122399, A229260.

Table[Sum[k^(3*n) * k! * StirlingS2[n,k], {k,0,n}], {n,0,20}]

a(n) = sum(k=0, n, k!*k^(3*n)*stirling(n, k, 2)); \\ Seiichi Manyama, Feb 01 2022

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (exp(k^3*x)-1)^k))) \\ Seiichi Manyama, Feb 01 2022

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!*(k^3*x)^k/prod(j=1, k, 1-k^3*j*x))) \\ Seiichi Manyama, Feb 01 2022

a(n) ~ c * d^n * (n!)^4 / n^(3/2), where d = 20.5647332000203822461493845960846630764635... = r^4*(1+exp(3/r)), r = 0.97762267432285928683132021521727105447350... is the root of the equation (1+exp(-3/r))*LambertW(-exp(-1/r)/r) = -1/r, and c = 0.0600744446309702764688382302731840300640714536...
E.g.f.: Sum_{k>=0} (exp(k^3*x) - 1)^k. - Seiichi Manyama, Feb 01 2022
G.f.: Sum_{k>=0} k! * (k^3*x)^k/Product_{j=1..k} (1 - k^3*j*x). - Seiichi Manyama, Feb 01 2022

a(0)=1 prepended by Seiichi Manyama, Feb 01 2022

A351137 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(3n) Stirling1(n,k).

Original entry on oeis.org

1, 1, 129, 121172, 421875178, 3922823960054, 80130334773241142, 3156849112458066440568, 218554371053209725986724984, 24795129220015277612148345850896, 4365539219231132131300647267518575008, 1141930521329052244894253748456776246166288

Offset: 0

Seiichi Manyama, Feb 02 2022

nonn

In general, for m >= 0, Sum_{k=0..n} (-1)^(n-k) * k! * k^(m*n) * Stirling1(n,k) ~ c * r^(m*n) * (1 + r*exp(m/r))^n * n^((m+1)*n + 1/2) / exp((m+1)*n), where r is the real root of the equation LambertW(-1, -r*exp(-r)) = -r - exp(-m/r) and c is a constant (depending only on m). - Vaclav Kotesovec, Feb 04 2022

Seiichi Manyama, Table of n, a(n) for n = 0..125

Cf. A007840 (m=0), A320096 (m=1), A351136 (m=2).

Cf. A203798, A351134, A351138.

a[0] = 1; a[n_] := Sum[(-1)^(n - k) * k! * k^(3*n) * StirlingS1[n, k], {k, 1, n}]; Array[a, 12, 0] (* Amiram Eldar, Feb 02 2022 *)

a(n) = sum(k=0, n, (-1)^(n-k)*k!*k^(3*n)*stirling(n, k, 1));

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-log(1-k^3*x))^k)))

E.g.f.: Sum_{k>=0} (-log(1 - k^3*x))^k.

a(n) ~ c * r^(3*n) * (1 + r*exp(3/r))^n * n^(4*n + 1/2) / exp(4*n), where r = 0.97698437755148201976772582981871258235824532360125531194... is the real root of the equation LambertW(-1, -r*exp(-r)) = -r - exp(-3/r) and c = 2.3655154360078103511101518906595610482889989819... - Vaclav Kotesovec, Feb 04 2022

A249584 E.g.f.: Sum_{n>=0} (1 - exp(-n^n*x))^n.

Original entry on oeis.org

1, 1, 31, 117715, 103060086811, 35762522984596014091, 7426384178532990386136937008571, 1294894823429942167077615079432957932604708795, 253092741940931724342360814670783323840910439695820558059377771, 71352024076415778396125047299738711969981288880155877082517115026490261985701403851

Offset: 0

Paul D. Hanna, Nov 01 2014

nonn

			E.g.f.: A(x) = 1 + x + 31*x^2/2! + 117715*x^3/3! + 103060086811*x^4/4! +...
where
A(x) = 1 + (1-exp(-x)) + (1-exp(-2^2*x))^2 + (1-exp(-3^3*x))^3 + (1-exp(-4^4*x))^4 + (1-exp(-5^5*x))^5 + (1-exp(-6^6*x))^6 +...
Ordinary generating function.
O.g.f.: F(x) = 1 + x + 31*x^2 + 117715*x^3 + 103060086811*x^4 +...
where
F(x) = 1 + x/(1+x) + 2^4*2!*x^2/((1+2^2*1*x)*(1+2^2*2*x)) + 3^9*3!*x^3/((1+3^3*1*x)*(1+3^3*2*x)*(1+3^3*3*x)) + 4^16*4!*x^4/((1+4^4*1*x)*(1+4^4*2*x)*(1+4^4*3*x)*(1+4^4*4*x)) + 5^25*5!*x^5/((1+5^5*1*x)*(1+5^5*2*x)*(1+5^5*3*x)*(1+5^5*4*x)*(1+5^5*5*x)) +...

Seiichi Manyama, Table of n, a(n) for n = 0..26

Cf. A220181, A187755, A203798.

Flatten[{1, Table[Sum[(-1)^(n-k)*(k^k)^n*k!*StirlingS2[n, k], {k, 1, n}], {n, 1, 10}]}] (* Vaclav Kotesovec, Aug 24 2017 *)

{a(n)=n!*polcoeff(sum(k=0, n, (1-exp(-k^k*x +x*O(x^n)))^k), n)}
for(n=0, 15, print1(a(n), ", "))

{a(n)=polcoeff(sum(m=0, n, m^(m^2)*m!*x^m/prod(k=1, m, 1+m^m*k*x +x*O(x^n))), n)}
for(n=0, 15, print1(a(n), ", "))

{a(n) = sum(k=0, n, (-1)^(n-k)*k^(k*n)*k!*stirling(n, k, 2))}
for(n=0, 10, print1(a(n), ", "))

a(n) == 1 (mod 6) for n>=0.
O.g.f.: Sum_{n>=0} n^(n^2) * n! * x^n / Product_{k=1..n} (1 + n^n*k*x).
a(n) = Sum_{k=0..n} (-1)^(n-k) * (k^k)^n * k! * Stirling2(n,k).
a(n) ~ n! * n^(n^2). - Vaclav Kotesovec, Aug 24 2017

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

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A187755 E.g.f.: Sum_{n>=0} (1 - exp(-n^2*x))^n.

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A242229 a(n) = Sum_{k=0..n} k^(3*n) * k! * StirlingS2(n,k).

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A351137 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(3*n) * Stirling1(n,k).

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A249584 E.g.f.: Sum_{n>=0} (1 - exp(-n^n*x))^n.

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A242229 a(n) = Sum_{k=0..n} k^(3n) k! * StirlingS2(n,k).

A351137 a(n) = Sum_{k=0..n} (-1)^(n-k) * k! * k^(3n) Stirling1(n,k).