A217913 -id:A217913 - cp's OEIS frontend

A007820 Stirling numbers of second kind S(2n,n).

1, 1, 7, 90, 1701, 42525, 1323652, 49329280, 2141764053, 106175395755, 5917584964655, 366282500870286, 24930204590758260, 1850568574253550060, 148782988064375309400, 12879868072770626040000, 1194461517469807833782085, 118144018577011378596484455

Offset: 0

kemp(AT)sads.informatik.uni-frankfurt.de (Rainer Kemp)

Chan and Manna prove that a(n) is odd if and only if n is in A003714. - Jason Kimberley, Sep 14 2009
The number of ways to partition a set of 2*n elements into n disjoint subsets. - Vladimir Reshetnikov, Oct 10 2016
Conjecture: a(2*n+1) is divisible by (2*n + 1)^2. - Peter Bala, Mar 30 2025

			G.f.: A(x) = 1 + x + 7*x^2 + 90*x^3 + 1701*x^4 + 42525*x^5 +...,
where A(x) = 1 + 1^2*x*exp(-1*x) + 2^4*exp(-2^2*x)*x^2/2! + 3^6*exp(-3^2*x)*x^3/3! + 4^8*exp(-4^2*x)*x^4/4! + 5^10*exp(-5^2*x)*x^5/5! + ... - _Paul D. Hanna_, Oct 17 2012

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.

Alois P. Heinz, Table of n, a(n) for n = 0..345 (terms n = 1..200 from Vincenzo Librandi)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
O-Y. Chan and D. V. Manna, Divisibility properties of Stirling numbers of the second kind [From _Jason Kimberley_, Sep 14 2009]

Cf. A002465, A008277, A187646, A191236, A217913, A217914, A217915, A247238.

Cf. A350376, A383862, A384060.

A007820 := proc(n) Stirling2(2*n,n) ; end proc:
seq(A007820(n),n=0..20) ; # R. J. Mathar, Mar 15 2011

Table[StirlingS2[2n, n], {n, 1, 12}] (* Emanuele Munarini, Mar 12 2011 *)

makelist(stirling2(2*n,n),n,0,12); /* Emanuele Munarini, Mar 12 2011 */

a(n)=stirling(2*n,n,2); /* Joerg Arndt, Jul 01 2011 */

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(2*n))), n)} \\ Paul D. Hanna, Oct 17 2012

{a(n)=polcoeff(sum(m=1,n,(m^2)^m*exp(-m^2*x+x*O(x^n))*x^m/m!),n)} \\ Paul D. Hanna, Oct 17 2012

from sympy.functions.combinatorial.numbers import stirling
def A007820(n): return stirling(n<<1,n) # Chai Wah Wu, Jun 09 2025

[stirling_number2(2*i,i) for i in range(1,20)] # Zerinvary Lajos, Jun 26 2008

a(n) = A048993(2n,n). - R. J. Mathar, Mar 15 2011
Asymptotic: a(n) ~ (4*n/(e*z*(2-z)))^n/sqrt(2*Pi*n*(z-1)), where z = A256500 = 1.59362426... is a root of the equation exp(z)*(2-z)=2. - Vaclav Kotesovec, May 30 2011
a(n) = 1/n! * Sum_{k = 0..n} binomial(n,k)*(-1)^k*(n-k)^(2*n). - Emanuele Munarini, Jul 01 2011
a(n) = [x^n] 1 / Product_{k=1..n} (1-k*x). - Paul D. Hanna, Oct 17 2012
O.g.f.: Sum_{n>=1} (n^2)^n * exp(-n^2*x) * x^n/n! = Sum_{n>=1} S2(2*n,n)*x^n. - Paul D. Hanna, Oct 17 2012
G.f.: Sum_{n > 0} (a(n)*n!/(2*n)!)*x^n = x*B'(x)/B(x)-1, where B(x) satisfies B(x)^2 = x*(exp(B(x))-1). - Vladimir Kruchinin, Mar 13 2013
a(n) = Sum_{j = 0..n} (-1)^(n-j)*n^j*binomial(2*n,j)*stirling2(2*n-j,n). - Vladimir Kruchinin, Jun 14 2013

Typo in Mathematica program fixed by Vincenzo Librandi, May 04 2013

a(0)=1 prepended by Alois P. Heinz, Feb 01 2018

A217900 O.g.f.: Sum_{n>=0} n^n * (n+1)^(n-1) * exp(-n(n+1)x) * x^n / n!.

Original entry on oeis.org

1, 1, 4, 38, 576, 12052, 322848, 10564304, 408903680, 18288706544, 928575662400, 52780935007968, 3321208845997056, 229232635832433664, 17221699990084108288, 1399139700462119135232, 122235936429355565580288, 11428226675376971405577984, 1138551595285580854471388160

Offset: 0

Paul D. Hanna, Oct 14 2012

nonn

Compare the g.f. to the LambertW identity:
1 = Sum_{n>=0} (n+1)^(n-1) * exp(-(n+1)*x) * x^n/n!.
More generally, if we define a(n) for fixed integers m, t, and s>=0, by:
(0) Sum_{n>=0} m * n^(s*n) * (n*t+m)^(n-1) * exp(-n^s*(n*t+m)*x) * x^n/n! = Sum_{n>=0} a(n)*x^n
then the coefficients a(n) are integral and may be expressed by:
(1) a(n) = 1/n! * Sum_{k=0..n} m*(-1)^(n-k)*binomial(n,k) * k^(s*n) * (k*t+m)^(n-1).
(2) a(n) = 1/n! * [x^n] Sum_{k>=0} m*k^(s*k)*(k*t+m)^(k-1)*x^k / (1 + k^s*(k*t+m)*x)^(k+1).
(3) a(n) = 1/t^((s-1)*n) * [x^(s*n)] 1 + m*x*(1+m*x)^(n-1) / Product_{k=1..n} (1-k*t*x).
(4) a(n) = 1/t^((s-1)*n) * [x^(s*n)] 1 + m*x*(1-m*x)^(s*n) / Product_{k=1..n} (1-(k*t+m)*x).

			O.g.f.: A(x) = 1 + x + 4*x^2 + 38*x^3 + 576*x^4 + 12052*x^5 + 322848*x^6 +...
where
A(x) = 1 + 1^1*2^0*x*exp(-1*2*x) + 2^2*3^1*exp(-2*3*x)*x^2/2! + 3^3*4^2*exp(-3*4*x)*x^3/3! + 4^4*5^3*exp(-4*5*x)*x^4/4! + 5^5*6^4*exp(-5*6*x)*x^5/5! +...
simplifies to a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..345

Cf. A078739, A217899, A217901, A217902, A217903, A217904, A217905, A217910, A217913, A218300.

a[n_] := 1/n!*Sum[(-1)^(n-k)*Binomial[n, k]*k^n*(k+1)^(n-1), {k, 0, n}]; a[0]=1; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Mar 06 2013 *)

{a(n)=polcoeff(sum(m=0,n,m^m*(m+1)^(m-1)*x^m*exp(-m*(m+1)*x+x*O(x^n))/m!),n)}

{a(n)=(1/n!)*polcoeff(sum(k=0, n, k^k*(k+1)^(k-1)*x^k/(1+k*(k+1)*x +x*O(x^n))^(k+1)), n)}

{a(n)=1/n!*sum(k=0,n, (-1)^(n-k)*binomial(n,k)*k^n*(k+1)^(n-1))}

{a(n)=polcoeff(1+x*(1+x)^(n-1)/prod(k=0, n, 1-k*x +x*O(x^n)), n)}

{a(n)=polcoeff(1+x*(1-x)^n/prod(k=0, n, 1-(k+1)*x +x*O(x^n)), n)}
for(n=0,30,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=0,n-1, binomial(n-1,k) * Stirling2(2*n-k-1,n)))} \\ Paul D. Hanna, Nov 13 2012
/* PARI Programs for the General Case (START) ...................... */

{a(n,m=1,t=1,s=1)=polcoeff(sum(k=0, n, m*k^(s*k)*(t*k+m)^(k-1)*exp(-k^s*(t*k+m)*x+x*O(x^n))*x^k/k!), n)}

{a(n,m=1,t=1,s=1)=(1/n!)*polcoeff(sum(k=0, n, m*k^(s*k)*(t*k+m)^(k-1)*x^k/(1+k^s*(t*k+m)*x +x*O(x^n))^(k+1)), n)}

{a(n,m=1,t=1,s=1)=1/n!*sum(k=0, n, m*(-1)^(n-k)*binomial(n, k)*k^(s*n)*(t*k+m)^(n-1))}

{a(n,m=1,t=1,s=1)=(1/t^((s-1)*n))*polcoeff(1+m*x*(1+m*x)^(n-1)/prod(k=0, n, 1-t*k*x +x*O(x^(s*n))), s*n)}

{a(n,m=1,t=1,s=1)=(1/t^((s-1)*n))*polcoeff(1+m*x*(1-m*x)^(s*n)/prod(k=0, n, 1-(t*k+m)*x +x*O(x^(s*n))), s*n)}
/* (END) ........................................................... */

a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (k+1)^(n-1).
a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(k+1)^(k-1)*x^k / (1 + k*(k+1)*x)^(k+1).
a(n) = [x^n] 1 + x*(1+x)^(n-1) / Product_{k=1..n} (1-k*x).
a(n) = [x^n] 1 + x*(1-x)^(n-1) / Product_{k=1..n} (1-(k+1)*x).
a(n) = A078739(n,n) for n>=1.
a(n) = Sum_{k=0..n-1} binomial(n-1,k) * Stirling2(2*n-k-1,n) for n>0, where Stirling2(n,k) = A008277(n,k). - Paul D. Hanna, Nov 13 2012
a(n) ~ 2^(2*n-1) * n^(n-3/2) / (sqrt(Pi*(1-c)) * exp(n) * (2-c)^(n-1) * c^(n+1/2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... = 2*A106533. - Vaclav Kotesovec, May 09 2014

A226750 Decimal expansion of the number x other than -3 defined by x*e^x = -3/e^3.

Original entry on oeis.org

1, 7, 8, 5, 6, 0, 6, 2, 7, 8, 7, 7, 9, 2, 1, 1, 0, 6, 5, 9, 6, 8, 0, 8, 6, 6, 9, 7, 0, 5, 5, 1, 4, 8, 0, 4, 6, 5, 4, 1, 1, 8, 2, 5, 5, 9, 2, 6, 8, 8, 5, 9, 0, 7, 7, 2, 0, 1, 4, 2, 3, 0, 6, 0, 5, 8, 7, 8, 5, 6, 9, 5, 4, 9, 4, 4, 8, 2, 6, 0, 8, 7, 5, 4, 3, 1, 3, 4, 3, 4, 6, 5, 2, 1, 6, 0, 3, 5, 5, 6, 1, 0, 4, 0, 0

Offset: 0

José María Grau Ribas, Jun 16 2013

			-0.178560627877921106596808669705514804654118255926885907720142306058785...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A106533, A217913, A221214, A274712, A274713.

RealDigits[N[ProductLog[-3/e^3], 105]][[1]]

solve(x=0,1,3/exp(3)-x*exp(-x)) \\ Charles R Greathouse IV, Nov 19 2013

Term a(104) corrected by G. C. Greubel, Nov 16 2017

A217914 O.g.f.: Sum_{n>=0} (n^4)^n * exp(-n^4x) x^n / n!.

Original entry on oeis.org

1, 1, 127, 86526, 171798901, 749206090500, 6090236036084530, 82892803728383735268, 1751346256720122175776157, 54294340536065700496358447625, 2364684125291482936353925428946680, 139762001313639974628848043262243505970, 10897986831117690497797320098390628446479030

Offset: 0

Paul D. Hanna, Oct 14 2012

nonn

			O.g.f.: A(x) = 1 + x + 127*x^2 + 86526*x^3 + 171798901*x^4 +...+ Stirling2(4*n,n)*x^n + ...
where
A(x) = 1 + 1^4*x*exp(-1^4*x) + 2^8*exp(-2^4*x)*x^2/2! + 3^12*exp(-3^4*x)*x^3/3! + 4^16*exp(-4^4*x)*x^4/4! + 5^20*exp(-5^4*x)*x^5/5! + ...
is a power series in x with integer coefficients.

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A007820, A217913, A217915, A217900, A008277.

Table[StirlingS2[4*n,n],{n,0,20}] (* Vaclav Kotesovec, May 23 2013 *)

makelist(stirling2(4*n, n), n, 0, 12); /* Martin Ettl, Oct 15 2012 */

{a(n)=polcoeff(sum(k=0,n,(k^4)^k*exp(-k^4*x +x*O(x^n))*x^k/k!),n)}

{a(n)=1/n!*polcoeff(sum(k=0, n, (k^4)^k*x^k/(1+k^4*x +x*O(x^n))^(k+1)), n)}

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(3*n))), 3*n)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(4*n, n)}
for(n=0,12,print1(a(n),", "))

a(n) = Stirling2(4*n, n).
a(n) = [x^(4*n)] (4*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(3*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^4)^k*x^k / (1 + k^4*x)^(k+1).
a(n) ~ 2^(8*n)*n^(3*n)/(sqrt(2*Pi*n*(1-c))*c^n*exp(3*n)*(4-c)^(3*n)), where c = -LambertW(-4/exp(4)) = 0.07930960512711... - Vaclav Kotesovec, May 23 2013

A217915 O.g.f.: Sum_{n>=1} (n^5)^n * exp(-n^5x) x^n / n!.

Original entry on oeis.org

1, 1, 511, 2375101, 45232115901, 2436684974110751, 299310102746948685757, 72786959006434393367186463, 31712979422428631132831124895809, 22982258052528294182955639980819773510, 26154716515862881292012777396577993781727011

Offset: 0

Paul D. Hanna, Oct 14 2012

nonn

			O.g.f.: A(x) = 1 + x + 511*x^2 + 2375101*x^3 + 45232115901*x^4 +...+ Stirling2(5*n, n)*x^n +...
where
A(x) = 1 + 1^5*x*exp(-1^5*x) + 2^10*exp(-2^5*x)*x^2/2! + 3^15*exp(-3^5*x)*x^3/3! + 4^20*exp(-4^5*x)*x^4/4! + 5^25*exp(-5^5*x)*x^5/5! +...
is a power series in x with integer coefficients.

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A007820, A217913, A217914, A217900, A008277.

Table[StirlingS2[5*n,n],{n,0,20}] (* Vaclav Kotesovec, May 23 2013 *)

makelist(stirling2(5*n, n), n, 0, 10); /* Martin Ettl, Oct 15 2012 */

{a(n)=polcoeff(sum(k=0,n,(k^5)^k*exp(-k^5*x +x*O(x^n))*x^k/k!),n)}

{a(n)=1/n!*polcoeff(sum(k=0, n, (k^5)^k*x^k/(1+k^5*x +x*O(x^n))^(k+1)), n)}

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(4*n))), 4*n)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(5*n, n)}
for(n=0,12,print1(a(n),", "))

a(n) = Stirling2(5*n, n).
a(n) = [x^(5*n)] (5*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(4*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^5)^k*x^k / (1 + k^5*x)^(k+1).
a(n) ~ n^(4*n)*5^(5*n) / (sqrt(2*Pi*n*(1-c)) * exp(4*n) * (5-c)^(4*n) * c^n), where c = -LambertW(-5/exp(5)) = 0.0348857682557... - Vaclav Kotesovec, May 23 2013

A129506 Number of partitions of a {2n-1}-set into n nonempty subsets.

Original entry on oeis.org

1, 3, 25, 350, 6951, 179487, 5715424, 216627840, 9528822303, 477297033785, 26826851689001, 1672162773483930, 114485073343744260, 8541149231801585700, 689692892575539953400, 59932861644880019603520, 5576731051262006158950735, 553234633385550257808059085

Offset: 1

Paul D. Hanna, Apr 18 2007

nonn

B^{-1}(x) = Sum_{n>0} a(n)/(2*n-1)!*(n-1)! x^n is inverse function for x*B(x), where B(x) is g.f. for Bernoulli number (see A027641). - Vladimir Kruchinin, Jan 19 2012

			G.f.: A(x) = x + 3*x^2 + 25*x^3 + 350*x^4 + 6951*x^5 + 179487*x^6 + ... where A(x) = 1^1*x*exp(-1^2*x) + 2^3*exp(-2^2*x)*x^2/2! + 3^5*exp(-3^2*x)*x^3/3! + 4^7*exp(-4^2*x)*x^4/4! + 5^9*exp(-5^2*x)*x^5/5! + ... forms a power series in x with integer coefficients. - _Paul D. Hanna_, Oct 15 2012

Alois P. Heinz, Table of n, a(n) for n = 1..200
D. Kruchinin and V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, Journal of Integer Sequences, Vol. 15 (2012), article 12.9.3.

Cf. A008277, A129505, A217900, A217910, A217913, A258170.

a:= n-> Stirling2(2*n-1, n):
seq(a(n), n=1..25);  # Alois P. Heinz, Dec 15 2013

a[n_] := Sum[ Binomial[2*n - 2, j]*StirlingS2[2*n - j - 2, n-1], {j, 0, n-1}]; Table[a[n], {n, 1, 18}] (* Jean-François Alcover, Jun 14 2013, after Vladimir Kruchinin *)
Table[StirlingS2[2*n-1,n], {n, 1, 20}] (* Vaclav Kotesovec, Dec 15 2013 *)

a(n):=((2*n-1)*((sum((stirling2(2*i-1, i)*binomial(2*n-2, 2*i-1)*stirling2(2*(n-i)-1, n-i-1))/((n-i-1)*binomial(n-1, i)), i, 1, n-2))+(n-1)* stirling2(2*n-3, n-1)+stirling2(2*n-2, n-1)))/(n);
  makelist(a(n),n,1,10); /* Vladimir Kruchinin, Feb 28 2013 */

a(n)=polcoeff(1/prod(k=1,n,1-k*x +x*O(x^n)),n-1)

vector(66, n, abs( stirling(2*n-1, n, 2) ) ) /* Joerg Arndt, Jun 09 2012 */

{a(n)=1/n!*sum(k=0,n, (-1)^(n-k)*binomial(n,k)*k^(2*n-1))} \\ Paul D. Hanna, Oct 15 2012

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

Central Stirling numbers of the second kind: a(n) = A008277(2n-1,n) for n >= 1.
G.f.: Sum_{n>=1} n^(2*n-1) * exp(-n^2*x) * x^n / n!, an integer series. - Paul D. Hanna, Oct 15 2012
a(n) = 1/n! * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * k^(2*n-1). - Paul D. Hanna, Oct 15 2012
a(n) = ((2*n-1)*((sum(i=1..n-2, (stirling2(2*i-1,i)*C(2*n-2,2*i-1)*stirling2(2*(n-i)-1,n-i-1))/((n-i-1)*C(n-1,i))))+(n-1)*stirling2(2*n-3,n-1) +stirling2(2*n-2,n-1)))/n. - Vladimir Kruchinin, Feb 28 2013
a(n-1) = sum(j=0..n, binomial(2*n,j)*stirling2(2*n-j,n)). - Vladimir Kruchinin, Jun 14 2013
a(n) ~ 2^(2*n-3/2) * n^(n-3/2) * (2-c)^(1-n) / (sqrt(Pi*(1-c)) * exp(n) * c^n), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... = 2*A106533. - Vaclav Kotesovec, Dec 15 2013
a(n) = A258170(2*n-1,n). - Alois P. Heinz, Mar 16 2018

A218141 a(n) = Stirling2(n^2, n).

Original entry on oeis.org

1, 1, 7, 3025, 171798901, 2436684974110751, 14204422416132896951197888, 50789872166903636182659702516635946082, 155440114706926165785630654089245708839702615196926765, 541500903058656141876322139677626107784896646583041951351456223689104719

Offset: 0

Paul D. Hanna, Oct 21 2012

nonn

			O.g.f.: A(x) = 1 + x + 7*x^2 + 3025*x^3 + 171798901*x^4 + 2436684974110751*x^5 +...

Paul D. Hanna, Table of n, a(n) for n = 0..25

Cf. A008277, A218142, A218143, A007820, A217913, A217914, A217915.

Table[StirlingS2[n^2, n],{n,0,10}] (* Vaclav Kotesovec, May 11 2014 *)

makelist(stirling2(n^2,n),n,0,30 ); /* Martin Ettl, Oct 21 2012 */

{a(n)=polcoeff(sum(k=0,n,(k^n)^k*exp(-k^n*x +x*O(x^n))*x^k/k!),n)}

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(n^2+1))), n^2-n)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(n^2, n)}
for(n=0, 10, print1(a(n), ", "))

a(n) = [x^n] Sum_{k>=0} k^(n*k) * exp(-k^n*x) * x^k / k!.
a(n) = [x^(n^2-n)] 1 / Product_{k=1..n} (1-k*x).
a(n) ~ n^(n^2)/n!. - Vaclav Kotesovec, May 11 2014

A222526 O.g.f.: Sum_{n>=0} (n^6)^n * exp(-n^6x) x^n / n!.

Original entry on oeis.org

1, 1, 2047, 64439010, 11681056634501, 7713000216608565075, 14204422416132896951197888, 61232072982330045410678351728440, 545827051514425992551826008968173372261, 9173647538352903119028122246836507680995590680

Offset: 0

Paul D. Hanna, Feb 23 2013

nonn

			O.g.f.: A(x) = 1 + x + 2047*x^2 + 64439010*x^3 + 11681056634501*x^4 +...+ Stirling2(6*n, n)*x^n +...
where
A(x) = 1 + 1^6*x*exp(-1^6*x) + 2^12*exp(-2^6*x)*x^2/2! + 3^18*exp(-3^6*x)*x^3/3! + 4^24*exp(-4^6*x)*x^4/4! + 5^30*exp(-5^6*x)*x^5/5! +...
is a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..96

Cf. A007820, A217913, A217914, A217915, A222527, A222528, A222529, A222530, A217900.

Table[StirlingS2[6*n, n],{n,0,20}] (* Vaclav Kotesovec, May 11 2014 *)

{a(n)=polcoeff(sum(k=0, n, (k^6)^k*exp(-k^6*x +x*O(x^n))*x^k/k!), n)}

{a(n)=1/n!*polcoeff(sum(k=0, n, (k^6)^k*x^k/(1+k^6*x +x*O(x^n))^(k+1)), n)}

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(5*n))), 5*n)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(6*n, n)}
for(n=0, 12, print1(a(n), ", "))

a(n) = Stirling2(6*n, n).
a(n) = [x^(6*n)] (6*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(5*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^6)^k*x^k / (1 + k^6*x)^(k+1).
a(n) ~ n^(5*n) * 6^(6*n) / (sqrt(2*Pi*(1-c)*n) * exp(5*n) * (6-c)^(5*n) * c^n), where c = -LambertW(-6*exp(-6)). - Vaclav Kotesovec, May 11 2014

A222527 O.g.f.: Sum_{n>=0} (n^7)^n * exp(-n^7x) x^n / n!.

Original entry on oeis.org

1, 1, 8191, 1742343625, 2998587019946701, 24204004899040755811870, 666480349285726891499539272955, 50789872166903636182659702516635946082, 9237419992097529135737293866043969707761346313, 3590622358224471993651445012122431990834934483552661750

Offset: 0

Paul D. Hanna, Feb 23 2013

nonn

			O.g.f.: A(x) = 1 + x + 8191*x^2 + 1742343625*x^3 + 2998587019946701*x^4 +...+ Stirling2(7*n, n)*x^n +...
where
A(x) = 1 + 1^7*x*exp(-1^7*x) + 2^14*exp(-2^7*x)*x^2/2! + 3^21*exp(-3^7*x)*x^3/3! + 4^28*exp(-4^7*x)*x^4/4! + 5^35*exp(-5^7*x)*x^5/5! +...
is a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..83

Cf. A007820, A217913, A217914, A217915, A222526, A222528, A222529, A222530, A217900.

Table[StirlingS2[7*n, n],{n,0,20}] (* Vaclav Kotesovec, May 11 2014 *)

{a(n)=polcoeff(sum(k=0, n, (k^7)^k*exp(-k^7*x +x*O(x^n))*x^k/k!), n)}

{a(n)=1/n!*polcoeff(sum(k=0, n, (k^7)^k*x^k/(1+k^7*x +x*O(x^n))^(k+1)), n)}

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(6*n))), 6*n)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(7*n, n)}
for(n=0, 12, print1(a(n), ", "))

a(n) = Stirling2(7*n, n).
a(n) = [x^(7*n)] (7*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(6*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^7)^k*x^k / (1 + k^7*x)^(k+1).
a(n) ~ n^(6*n) * 7^(7*n) / (sqrt(2*Pi*(1-c)*n) * exp(6*n) * (7-c)^(6*n) * c^n), where c = -LambertW(-7*exp(-7)). - Vaclav Kotesovec, May 11 2014

A222528 O.g.f.: Sum_{n>=0} (n^8)^n * exp(-n^8x) x^n / n!.

Original entry on oeis.org

1, 1, 32767, 47063200806, 768305500780164501, 75740854251732106906082250, 31154086963475828638359480518580526, 41929298560838945526242744414099901692285884, 155440114706926165785630654089245708839702615196926765, 1396002062838446082394548660243302585983358463911636390911298400

Offset: 0

Paul D. Hanna, Feb 23 2013

nonn

			O.g.f.: A(x) = 1 + x + 32767*x^2 + 47063200806*x^3 + 768305500780164501*x^4 +...+ Stirling2(8*n, n)*x^n +...
where
A(x) = 1 + 1^8*x*exp(-1^8*x) + 2^16*exp(-2^8*x)*x^2/2! + 3^24*exp(-3^8*x)*x^3/3! + 4^32*exp(-4^8*x)*x^4/4! + 5^40*exp(-5^8*x)*x^5/5! +...
is a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 0..100

Cf. A007820, A217913, A217914, A217915, A222526, A222527, A222529, A222530, A217900.

Table[StirlingS2[8*n, n],{n,0,20}] (* Vaclav Kotesovec, May 11 2014 *)

{a(n)=polcoeff(sum(k=0, n, (k^8)^k*exp(-k^8*x +x*O(x^n))*x^k/k!), n)}

{a(n)=1/n!*polcoeff(sum(k=0, n, (k^8)^k*x^k/(1+k^8*x +x*O(x^n))^(k+1)), n)}

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(7*n))), 7*n)}

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = Stirling2(8*n, n)}
for(n=0, 12, print1(a(n), ", "))

a(n) = Stirling2(8*n, n).
a(n) = [x^(8*n)] (8*n)! * (exp(x) - 1)^n / n!.
a(n) = [x^(7*n)] 1 / Product_{k=1..n} (1-k*x).
a(n) = 1/n! * [x^n] Sum_{k>=0} (k^8)^k*x^k / (1 + k^8*x)^(k+1).
a(n) ~ n^(7*n) * 8^(8*n) / (sqrt(2*Pi*(1-c)*n) * exp(7*n) * (8-c)^(7*n) * c^n), where c = -LambertW(-8*exp(-8)). - Vaclav Kotesovec, May 11 2014

cp's OEIS Frontend

A007820 Stirling numbers of second kind S(2n,n).

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A217900 O.g.f.: Sum_{n>=0} n^n * (n+1)^(n-1) * exp(-n*(n+1)*x) * x^n / n!.

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A226750 Decimal expansion of the number x other than -3 defined by x*e^x = -3/e^3.

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A217914 O.g.f.: Sum_{n>=0} (n^4)^n * exp(-n^4*x) * x^n / n!.

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A217915 O.g.f.: Sum_{n>=1} (n^5)^n * exp(-n^5*x) * x^n / n!.

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A217900 O.g.f.: Sum_{n>=0} n^n * (n+1)^(n-1) * exp(-n(n+1)x) * x^n / n!.

A217914 O.g.f.: Sum_{n>=0} (n^4)^n * exp(-n^4x) x^n / n!.

A217915 O.g.f.: Sum_{n>=1} (n^5)^n * exp(-n^5x) x^n / n!.

A222526 O.g.f.: Sum_{n>=0} (n^6)^n * exp(-n^6x) x^n / n!.

A222527 O.g.f.: Sum_{n>=0} (n^7)^n * exp(-n^7x) x^n / n!.

A222528 O.g.f.: Sum_{n>=0} (n^8)^n * exp(-n^8x) x^n / n!.