A217910 -id:A217910 - cp's OEIS frontend

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

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

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

A106533 The rumor constant: decimal expansion of the number x defined by xe^(2 - 2x) = 1.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, May 03 2005

			c = 0.20318786997997995383847906206241987910549878759057031750024774...

G. C. Greubel, Table of n, a(n) for n = 0..5000
L. Bayon, P. Fortuny Ayuso, J. M. Grau, A. M. Oller-Marcen, M. M. Ruiz, The Best-or-Worst and the Postdoc problems, arXiv:1706.07185 [math.PR], 2017.
L. Bayon, J. Grau, A. M. Oller-Marcen, M. Ruiz, P. M. Suarez, A variant of the Secretary Problem: the Best or the Worst, arXiv preprint arXiv:1603.03928 [math.PR], 2016.
D. J. Daley and D. G. Kendall, Stochastic rumours, Journal of the Institute of Mathematics and Its Applications 1:42-55, 1965.
Steven R. Finch, Errata and Addenda to Mathematical Constants.
José María Grau Ribas, A New Proof and Extension of the Odds-Theorem, arXiv:1812.09255 [math.PR], 2018.
José María Grau Ribas, An extension of the Last-Success-Problem, Statistics & Probability Letters (2020) Vol. 156, 108591.
Brian Hayes, Rumours and Errours, American Scientist, Vol. 93, Nbr. 3, 2005, pp. 207-211.
Index entries for transcendental numbers.

Cf. A007456, A129506, A217900, A217910, A256500.

RealDigits[ -ProductLog[ -2/E^2]/2, 10, 111][[1]]
RealDigits[x/.FindRoot[x E^(2-2x)==1,{x,2},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Jul 05 2025 *)

solve(x=0, 0.5, x*exp(2-2*x)-1) \\ Michel Marcus, Mar 13 2016

Solution to x*exp(2 - 2*x) = 1 with x not equal to 1.
Equals -1/2*LambertW(-2*exp(-2)). - Vladeta Jovovic, May 30 2005
Constant c satisfies: exp(c*x)/(1-2*c) = Sum_{n>=0} (x + 2*n)^n * exp(-2*n)/n!. - Paul D. Hanna, Mar 12 2016
Equals (2-A256500)/2. - Miko Labalan, Dec 18 2024

A217902 O.g.f.: Sum_{n>=0} 3n^n(n+3)^(n-1) * exp(-n(n+3)x) * x^n / n!.

Original entry on oeis.org

1, 3, 18, 210, 3696, 86436, 2521800, 88274640, 3608360064, 168822613872, 8901871248480, 522534101560224, 33804242536287744, 2390169742849449216, 183412961210465667072, 15183107016739655860224, 1348837954231568133427200, 128012762381954718934183680

Offset: 0

Paul D. Hanna, Oct 14 2012

nonn

Compare the g.f. to the LambertW identity:

1 = Sum_{n>=0} 3*(n+3)^(n-1) * exp(-(n+3)*x) * x^n/n!.

			O.g.f.: A(x) = 1 + 3*x + 18*x^2 + 210*x^3 + 3696*x^4 + 86436*x^5 + 2521800*x^6 +...
where
A(x) = 1 + 3*1^1*4^0*x*exp(-1*4*x) + 3*2^2*5^1*exp(-2*5*x)*x^2/2! + 3*3^3*6^2*exp(-3*6*x)*x^3/3! + 3*4^4*7^3*exp(-4*7*x)*x^4/4! + 3*5^5*8^4*exp(-5*8*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..250

Cf. A217899, A217900, A217901, A217903, A217904, A217905, A217910.

Flatten[{1,Table[Sum[Binomial[n-1,j]*3^(n-j)*StirlingS2[n+j,n],{j,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, May 22 2014 *)

{a(n)=polcoeff(sum(m=0,n,3*m^m*(m+3)^(m-1)*x^m*exp(-m*(m+3)*x+x*O(x^n))/m!),n)}

{a(n)=(1/n!)*polcoeff(sum(k=0, n, 3*k^k*(k+3)^(k-1)*x^k/(1+k*(k+3)*x +x*O(x^n))^(k+1)), n)}

{a(n)=1/n!*sum(k=0,n, 3*(-1)^(n-k)*binomial(n,k)*k^n*(k+3)^(n-1))}

{a(n)=polcoeff(1+3*x*(1+3*x)^(n-1)/prod(k=0, n, 1-k*x +x*O(x^n)), n)}

{a(n)=polcoeff(1+3*x*(1-3*x)^n/prod(k=0, n, 1-(k+3)*x +x*O(x^n)), n)}
for(n=0,30,print1(a(n),", "))

a(n) = 1/n! * Sum_{k=0..n} 3*(-1)^(n-k)*binomial(n,k) * k^n * (k+3)^(n-1).
a(n) = 1/n! * [x^n] Sum_{k>=0} 3*k^k*(k+3)^(k-1)*x^k / (1 + k*(k+3)*x)^(k+1).
a(n) = [x^n] 1 + 3*x*(1+3*x)^(n-1) / Product_{k=1..n} (1-k*x).
a(n) = [x^n] 1 + 3*x*(1-3*x)^(n-1) / Product_{k=1..n} (1-(k+3)*x).
a(n) ~ 3 * 2^(2*n) * n^(n-3/2) / (sqrt(Pi*(1-c)) * exp(n) * (2-c)^(n-1) * c^(n+3/2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 22 2014

A217903 O.g.f.: Sum_{n>=0} 4n^n(n+4)^(n-1) * exp(-n(n+4)x) * x^n / n!.

Original entry on oeis.org

1, 4, 28, 356, 6696, 165148, 5030124, 182425664, 7681137152, 368519318396, 19855601635860, 1187545259985444, 78096484084586904, 5602487847925307152, 435490669526307321808, 36468662242145922271968, 3273635846285796437437824, 313622489632532976209812284

Offset: 0

Paul D. Hanna, Oct 14 2012

nonn

Compare the g.f. to the LambertW identity:

1 = Sum_{n>=0} 4*(n+4)^(n-1) * exp(-(n+4)*x) * x^n/n!.

			O.g.f.: A(x) = 1 + 4*x + 28*x^2 + 356*x^3 + 6696*x^4 + 165148*x^5 + 5030124*x^6 +...
where
A(x) = 1 + 4*1^1*5^0*x*exp(-1*5*x) + 4*2^2*6^1*exp(-2*6*x)*x^2/2! + 4*3^3*7^2*exp(-3*7*x)*x^3/3! + 4*4^4*8^3*exp(-4*8*x)*x^4/4! + 4*5^5*9^4*exp(-5*9*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..250

Cf. A217899, A217900, A217901, A217902, A217904, A217905, A217910.

Flatten[{1,Table[Sum[Binomial[n-1,j]*4^(n-j)*StirlingS2[n+j,n],{j,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, May 22 2014 *)

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

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

{a(n)=1/n!*sum(k=0,n, 4*(-1)^(n-k)*binomial(n,k)*k^n*(k+4)^(n-1))}

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

{a(n)=polcoeff(1+4*x*(1-4*x)^n/prod(k=0, n, 1-(k+4)*x +x*O(x^n)), n)}
for(n=0,30,print1(a(n),", "))

a(n) = 1/n! * Sum_{k=0..n} 4*(-1)^(n-k)*binomial(n,k) * k^n * (k+4)^(n-1).
a(n) = 1/n! * [x^n] Sum_{k>=0} 4*k^k*(k+4)^(k-1)*x^k / (1 + k*(k+4)*x)^(k+1).
a(n) = [x^n] 1 + 4*x*(1+4*x)^(n-1) / Product_{k=1..n} (1-k*x).
a(n) = [x^n] 1 + 4*x*(1-4*x)^(n-1) / Product_{k=1..n} (1-(k+4)*x).
a(n) ~ 2^(2*n+5/2) * n^(n-3/2) / (sqrt(Pi*(1-c)) * exp(n) * (2-c)^(n-1) * c^(n+2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 22 2014

A217911 O.g.f.: Sum_{n>=0} n^n * (3n+1)^(n-1) exp(-n(3n+1)x) x^n / n!.

Original entry on oeis.org

1, 1, 10, 262, 11296, 684172, 53598952, 5162269744, 590636585728, 78321222303184, 11815503098606560, 1998732510370890208, 374763163567227915520, 77151431783218955979520, 17301697176590720940003328, 4198491769695976346962419712, 1096165878182404669364316147712

Offset: 0

Paul D. Hanna, Oct 14 2012

nonn

Compare the g.f. to the LambertW identity:

1 = Sum_{n>=0} (3*n+1)^(n-1) * exp(-(3*n+1)*x) * x^n/n!.

			O.g.f.: A(x) = 1 + x + 10*x^2 + 262*x^3 + 11296*x^4 + 684172*x^5 +...
where
A(x) = 1 + 1^1*4^0*x*exp(-1*4*x) + 2^2*7^1*exp(-2*7*x)*x^2/2! + 3^3*10^2*exp(-3*10*x)*x^3/3! + 4^4*13^3*exp(-4*13*x)*x^4/4! + 5^5*16^4*exp(-5*16*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..300

Cf. A217900, A217910, A217912.

Flatten[{1,Table[Sum[Binomial[n-1,j]*3^j*StirlingS2[n+j,n],{j,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, May 22 2014 *)

{a(n)=polcoeff(sum(k=0,n,k^k*(3*k+1)^(k-1)*x^k*exp(-k*(3*k+1)*x+x*O(x^n))/k!),n)}
for(n=0,30,print1(a(n),", "))

{a(n)=1/n!*polcoeff(sum(k=0, n, k^k*(3*k+1)^(k-1)*x^k/(1+k*(3*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*(3*k+1)^(n-1))}
for(n=0,30,print1(a(n),", "))

{a(n)=polcoeff(1+x*(1+x)^(n-1)/prod(k=0, n, 1-3*k*x +x*O(x^n)), n)}
for(n=0,30,print1(a(n),", "))

{a(n)=polcoeff(1+x*(1-x)^n/prod(k=0, n, 1-(3*k+1)*x +x*O(x^n)), n)}
for(n=0,30,print1(a(n),", "))

a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (3*k+1)^(n-1).
a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(3*k+1)^(k-1)*x^k / (1 + k*(3*k+1)*x)^(k+1).
a(n) = [x^n] 1 + x*(1+x)^(n-1) / Product_{k=1..n} (1 - 3*k*x).
a(n) = [x^n] 1 + x*(1-x)^(n-1) / Product_{k=1..n} (1 - (3*k+1)*x).
a(n) ~ 2^(2*n-4/3) * 3^(n-1) * n^(n-3/2) / (sqrt(Pi*(1-c)) * exp(n) * (2-c)^(n-1) * c^(n+1/6)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 22 2014

A217912 O.g.f.: Sum_{n>=0} 2n^n (3n+2)^(n-1) exp(-n(3n+2)x) x^n / n!.

Original entry on oeis.org

1, 2, 22, 602, 26656, 1643054, 130318966, 12666846728, 1459524093232, 194626267782398, 29495119281572770, 5008297010070635978, 942044179147597185544, 194462342099815302424136, 43711609296992502659474632, 10628894996508864880841838416, 2780041837527932453797746700384

Offset: 0

Paul D. Hanna, Oct 14 2012

nonn

Compare the g.f. to the LambertW identity:

1 = Sum_{n>=0} 2*(3*n+2)^(n-1) * exp(-(3*n+2)*x) * x^n/n!.

			O.g.f.: A(x) = 1 + 2*x + 22*x^2 + 602*x^3 + 26656*x^4 + 1643054*x^5 + ...
where
A(x) = 1 + 2*1^1*5^0*x*exp(-1*5*x) + 2*2^2*8^1*exp(-2*8*x)*x^2/2! + 2*3^3*11^2*exp(-3*11*x)*x^3/3! + 2*4^4*14^3*exp(-4*14*x)*x^4/4! + 2*5^5*17^4*exp(-5*17*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..300

Cf. A217900, A217910, A217911, A217913.

Flatten[{1,Table[Sum[Binomial[n-1,j]*3^j*2^(n-j)*StirlingS2[n+j,n],{j,0,n-1}],{n,1,20}]}] (* Vaclav Kotesovec, May 22 2014 *)

makelist( if n=0 then 1 else 1/n! * sum(2*(-1)^(n-k)*binomial(n,k) * k^n * (3*k+2)^(n-1),k,0,n), n, 0, 30); /* Martin Ettl, Oct 15 2012 */

{a(n)=polcoeff(sum(k=0,n,2*k^k*(3*k+2)^(k-1)*x^k*exp(-k*(3*k+2)*x+x*O(x^n))/k!),n)}
for(n=0,30,print1(a(n),", "))

{a(n)=1/n!*polcoeff(sum(k=0, n, 2*k^k*(3*k+2)^(k-1)*x^k/(1+k*(3*k+2)*x +x*O(x^n))^(k+1)), n)}

{a(n)=1/n!*sum(k=0,n, 2*(-1)^(n-k)*binomial(n,k)*k^n*(3*k+2)^(n-1))}
for(n=0,30,print1(a(n),", "))

{a(n)=polcoeff(1+2*x*(1+2*x)^(n-1)/prod(k=0, n, 1-3*k*x +x*O(x^n)), n)}
for(n=0,30,print1(a(n),", "))

{a(n)=polcoeff(1+2*x*(1-2*x)^n/prod(k=0, n, 1-(3*k+2)*x +x*O(x^n)), n)}
for(n=0,30,print1(a(n),", "))

a(n) = 1/n! * Sum_{k=0..n} 2*(-1)^(n-k)*binomial(n,k) * k^n * (3*k+2)^(n-1).
a(n) = 1/n! * [x^n] Sum_{k>=0} 2*k^k*(3*k+2)^(k-1)*x^k / (1 + k*(3*k+2)*x)^(k+1).
a(n) = [x^n] 1 + x*(1+2*x)^(n-1) / Product_{k=1..n} (1 - 3*k*x).
a(n) = [x^n] 1 + x*(1-2*x)^(n-1) / Product_{k=1..n} (1 - (3*k+2)*x).
a(n) ~ 2^(2*n-1/6) * 3^(n-1) * n^(n-3/2) / (sqrt(Pi*(1-c)) * exp(n) * (2-c)^(n-1) * c^(n+1/3)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 22 2014

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