A217901 -id:A217901 - 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

A217899 O.g.f.: Sum_{n>=1} (n^2)^(n-1) * exp(-n^2x) x^n / n!.

Original entry on oeis.org

1, 1, 6, 65, 1050, 22827, 627396, 20912320, 820784250, 37112163803, 1900842429486, 108823356051137, 6888836057922000, 477898618396288260, 36060660300744309600, 2940812098256837097720, 257780560811305783833450, 24171700822696604400643035, 2414448376056191692970387250

Offset: 1

Paul D. Hanna, Oct 14 2012

nonn

For n>1, a(n) is the number of set partitions of [2*n-2] into n blocks, i.e., Stirling2(2*n-2, n). E.g., a(3) = 6: [12|3|4, 13|2|4, 1|23|4, 14|2|3, 1|24|3, 1|2|34]. - Yuchun Ji, Jan 12 2021

			O.g.f.: A(x) = x + x^2 + 6*x^3 + 65*x^4 + 1050*x^5 + 22827*x^6 + 627396*x^7 + ... where A(x) = 1^0*x*exp(-1*x) + 2^2*exp(-2^2*x)*x^2/2! + 3^4*exp(-3^2*x)*x^3/3! + 4^6*exp(-4^2*x)*x^4/4! + 5^8*exp(-5^2*x)*x^5/5! + ... simplifies to a power series in x with integer coefficients.

G. C. Greubel, Table of n, a(n) for n = 1..250

Cf. A217900, A217901, A217902, A217903, A217904, A217905.

a[n_] := Sum[ Binomial[2*n - 3, j]*StirlingS2[2*n - j - 3, n-1], {j, 0, n-2}]; a[1] = 1; Table[a[n], {n, 1, 19}] (* Jean-François Alcover, Jun 14 2013, after Vladimir Kruchinin *)

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

{a(n)=1/n!*sum(k=1,n, (-1)^(n-k)*binomial(n,k)*(k^2)^(n-1))}

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

{a(n)=polcoeff(x+x^2*(1+x)^(2*n-3)/prod(k=0, n-1, 1-k*x +x*O(x^n)), n)}
for(n=1,20,print1(a(n),", "))

a(n) = (1/n!) * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * (k^2)^(n-1).
a(n) = [x^n] x + x^2/Product_{k=1..n} (1-k*x).
a(n) = [x^n] x + x^2*(1+x)^(2*n-3) / Product_{k=1..n-1} (1-k*x).
a(n) = Sum_{j=0..n-1} binomial(2*n-1,j)*Stirling2(2*n-j-1,n). - Vladimir Kruchinin, Jun 14 2013
a(n) ~ 2^(2*n-5/2) * n^(n-5/2) / (sqrt(Pi*(1-c)) * exp(n) * c^n *(2-c)^(n-2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 20 2014

A217905 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, -2, -14, -184, -3532, -89256, -2800016, -104967808, -4578528464, -227816059360, -12735645181536, -790296855912576, -53905019035510528, -4008716449677965312, -322807879692969879552, -27983800239966141382656, -2598368754552749176202496, -257284990746988090769530368

Offset: 0

Paul D. Hanna, Oct 14 2012

sign

Compare the g.f. to the LambertW identity:

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

			O.g.f.: A(x) = 1 - x - 2*x^2 - 14*x^3 - 184*x^4 - 3532*x^5 - 89256*x^6 +...
where
A(x) = 1 - 1^1*0^0*x*exp(-1*0*x) - 2^2*1^1*exp(-2*1*x)*x^2/2! - 3^3*2^2*exp(-3*2*x)*x^3/3! - 4^4*3^3*exp(-4*3*x)*x^4/4! - 5^5*4^4*exp(-5*4*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. A191236, A217906, A217899, A217900, A217901, A217902, A217903, A217904.

Join[{1, -1}, Table[(1/n!)*Sum[(-1)^(n - k + 1)*Binomial[n, k]*k^n*(k - 1)^(n - 1), {k, 0, n}], {n, 2, 50}]] (* G. C. Greubel, Nov 16 2017 *)

{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),", "))

a(n) = -A191236(n-1) for n>=1. [corrected by Vaclav Kotesovec, Aug 22 2018]
a(n) = 1/n! * Sum_{k=0..n} -(-1)^(n-k)*binomial(n,k) * k^n * (k-1)^(n-1) for n>=0.
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) ~ -2^(n-1) * exp(n*(r-1)-r) * n^(n - 3/2) / (sqrt(Pi*(r-1)*(2-r)) * r^(n-1)), where r = 2 + LambertW(-2*exp(-2)) = A256500 = 1.5936242600400400923230418... - Vaclav Kotesovec, Aug 22 2018

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

A217904 O.g.f.: Sum_{n>=0} 5n^n(n+5)^(n-1) * exp(-n(n+5)x) * x^n / n!.

Original entry on oeis.org

1, 5, 40, 550, 11000, 285380, 9064560, 340521520, 14773539200, 727281054640, 40072285049600, 2444188361990880, 163550098793059200, 11915396563502988800, 939110495156447488000, 79629365649015094272000, 7229173136192077603737600, 699726658343948617515436800

Offset: 0

Paul D. Hanna, Oct 14 2012

nonn

Compare the g.f. to the LambertW identity:
1 = Sum_{n>=0} 5*(n+5)^(n-1) * exp(-(n+5)*x) * x^n/n!.
From Vaclav Kotesovec, May 22 2014: (Start)
Generally, for p>=1, a(n) = 1/n!*Sum_{k=0..n} p*(-1)^(n-k) * binomial(n,k) * k^n * (k+p)^(n-1) = Sum_{j=0..n-1} binomial(n-1,j) * p^(n-j) * StirlingS2(n+j,n).
a(n) ~ p * 2^(2*n-3/2+p/2) * n^(n-3/2) / (sqrt(Pi*(1-c)) * exp(n) * (2-c)^(n-1) * c^(n+p/2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599...
(End)

			O.g.f.: A(x) = 1 + 5*x + 40*x^2 + 550*x^3 + 11000*x^4 + 285380*x^5 + 9064560*x^6 +...
where
A(x) = 1 + 5*1^1*6^0*x*exp(-1*6*x) + 5*2^2*7^1*exp(-2*7*x)*x^2/2! + 5*3^3*8^2*exp(-3*8*x)*x^3/3! + 5*4^4*9^3*exp(-4*9*x)*x^4/4! + 5*5^5*10^4*exp(-5*10*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..340

Cf. A217899, A217900, A217901, A217902, A217903, A217905.

Flatten[{1,Table[Sum[Binomial[n-1,j]*5^(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,5*m^m*(m+5)^(m-1)*x^m*exp(-m*(m+5)*x+x*O(x^n))/m!),n)}

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

{a(n)=1/n!*sum(k=0,n, 5*(-1)^(n-k)*binomial(n,k)*k^n*(k+5)^(n-1))}

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

{a(n)=polcoeff(1+5*x*(1-5*x)^n/prod(k=0, n, 1-(k+5)*x +x*O(x^n)), n)}
for(n=0,30,print1(a(n),", "))

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

A217910 O.g.f.: Sum_{n>=0} n^n(2n+1)^(n-1) * exp(-n(2n+1)x) x^n / n!.

Original entry on oeis.org

1, 1, 7, 125, 3641, 148297, 7792275, 502572905, 38466067169, 3409770740129, 343687137315215, 38829855954523317, 4861184771611069929, 668044273723230765337, 99988042875734734075243, 16191529121372446646518737, 2820684538705808192370559425

Offset: 0

Paul D. Hanna, Oct 14 2012

nonn

Compare the g.f. to the LambertW identity:

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

			O.g.f.: A(x) = 1 + x + 7*x^2 + 125*x^3 + 3641*x^4 + 148297*x^5 + 7792275*x^6 +...
where
A(x) = 1 + 1^1*3^0*x*exp(-1*3*x) + 2^2*5^1*exp(-2*5*x)*x^2/2! + 3^3*7^2*exp(-3*7*x)*x^3/3! + 4^4*9^3*exp(-4*9*x)*x^4/4! + 5^5*11^4*exp(-5*11*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..315

Cf. A217899, A217900, A217901, A217902, A217903, A217905, A217911.

Flatten[{1,Table[Sum[Binomial[n-1,j]*2^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*(2*k+1)^(k-1)*x^k*exp(-k*(2*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*(2*k+1)^(k-1)*x^k/(1+k*(2*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*(2*k+1)^(n-1))}
for(n=0,30,print1(a(n),", "))

{a(n)=polcoeff(1+x*(1+x)^(n-1)/prod(k=0, n, 1-2*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-(2*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 * (2*k+1)^(n-1).
a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(2*k+1)^(k-1)*x^k / (1 + k*(2*k+1)*x)^(k+1).
a(n) = [x^n] 1 + x*(1+x)^(n-1) / Product_{k=1..n} (1 - 2*k*x).
a(n) = [x^n] 1 + x*(1-x)^(n-1) / Product_{k=1..n} (1 - (2*k+1)*x).
a(n) ~ 2^(3*n-9/4) * n^(n-3/2) / (sqrt(Pi*(1-c)) * exp(n) * (2-c)^(n-1) * c^(n+1/4)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... = 2*A106533. - Vaclav Kotesovec, May 22 2014

A222076 O.g.f.: Sum_{n>=0} n^n(n+2)^n exp(-n(n+2)x) * x^n / n!.

Original entry on oeis.org

1, 3, 23, 320, 6397, 166467, 5338412, 203578776, 9001795829, 452924585465, 25555585227999, 1598279794889076, 109748572718377660, 8209004345714098500, 664396187060996529528, 57853075421585981420208, 5393119810256349152565573, 535908449308064099732283429, 56548822143306498413322880709

Offset: 0

Paul D. Hanna, Feb 24 2013

nonn

			O.g.f.: A(x) = 1 + 3*x + 23*x^2 + 320*x^3 + 6397*x^4 + 166467*x^5 +...
where
A(x) = 1 + 3*x*exp(-3*x) + 8^2*exp(-8*x)*x^2/2! + 15^3*exp(-15*x)*x^3/3! + 24^4*exp(-24*x)*x^4/4! + 35^5*exp(-35*x)*x^5/5! +...
is a power series in x with integer coefficients.

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

Cf. A217901, A217900, A222077, A222078, A222079.

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

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

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

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

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

A222523 O.g.f.: Sum_{n>=0} (n^2+n+1)^n * exp(-(n^2+n+1)x) x^n / n!.

Original entry on oeis.org

1, 2, 16, 208, 3930, 97956, 3038968, 112911296, 4889301222, 241822567180, 13450863716048, 831128810329632, 56483233790927556, 4187162929534240488, 336244786874092579920, 29077531985735270053632, 2694076376135933879002566, 266245292488900189811625612, 27956094249950913890814701248

Offset: 0

Paul D. Hanna, Feb 24 2013

nonn

			O.g.f.: A(x) = 1 + 2*x + 16*x^2 + 208*x^3 + 3930*x^4 + 97956*x^5 +...
where
A(x) = exp(-x) + 3*x*exp(-3*x) + 7^2*exp(-7*x)*x^2/2! + 13^3*exp(-13*x)*x^3/3! + 21^4*exp(-21*x)*x^4/4! + 31^5*exp(-31*x)*x^5/5! +...
is a power series in x with integer coefficients.

Cf. A217900, A217901.

{a(n)=polcoeff(sum(k=0, n, (k^2+k+1)^k*exp(-(k^2+k+1)*x +x*O(x^n))*x^k/k!), n)}
for(n=0, 25, print1(a(n), ", "))

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

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

a(n) = 1/n! * [x^n] Sum_{k>=0} (k^2+k+1)^k * x^k / (1 + (k^2+k+1)*x)^(k+1).

a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * (k^2+k+1)^n.

A222524 O.g.f.: Sum_{n>=0} n^n(2n+1)^n * exp(-n(2n+1)x) x^n / n!.

Original entry on oeis.org

1, 3, 41, 1057, 40057, 2006631, 125093285, 9333786225, 811181004929, 80480710535035, 8975976702322401, 1111688368710017121, 151388120776146737641, 22482576760232188394991, 3616177985990080869347277, 626250139757797928093888481, 116181112230230754285955844865

Offset: 0

Paul D. Hanna, Feb 24 2013

nonn

			O.g.f.: A(x) = 1 + 3*x + 41*x^2 + 1057*x^3 + 40057*x^4 + 2006631*x^5 +...
where
A(x) = 1 + 3*x*exp(-3*x) + 10^2*exp(-10*x)*x^2/2! + 21^3*exp(-21*x)*x^3/3! + 36^4*exp(-36*x)*x^4/4! + 55^5*exp(-55*x)*x^5/5! +...
is a power series in x with integer coefficients.

Cf. A217900, A217901.

{a(n)=polcoeff(sum(k=0, n, k^k*(2*k+1)^k*exp(-k*(2*k+1)*x +x*O(x^n))*x^k/k!), n)}
for(n=0, 25, print1(a(n), ", "))

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

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

a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(2*k+1)^k * x^k / (1 + k*(2*k+1)*x)^(k+1).

a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n*(2*k+1)^n.

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!.

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A217899 O.g.f.: Sum_{n>=1} (n^2)^(n-1) * exp(-n^2*x) * x^n / n!.

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A217905 O.g.f.: Sum_{n>=0} -n^n*(n-1)^(n-1) * exp(-n*(n-1)*x) * x^n / n!.

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

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

A217899 O.g.f.: Sum_{n>=1} (n^2)^(n-1) * exp(-n^2x) x^n / n!.

A217905 O.g.f.: Sum_{n>=0} -n^n(n-1)^(n-1) exp(-n(n-1)x) * x^n / n!.

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

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

A217904 O.g.f.: Sum_{n>=0} 5n^n(n+5)^(n-1) * exp(-n(n+5)x) * x^n / n!.

A217910 O.g.f.: Sum_{n>=0} n^n(2n+1)^(n-1) * exp(-n(2n+1)x) x^n / n!.

A222076 O.g.f.: Sum_{n>=0} n^n(n+2)^n exp(-n(n+2)x) * x^n / n!.

A222523 O.g.f.: Sum_{n>=0} (n^2+n+1)^n * exp(-(n^2+n+1)x) x^n / n!.

A222524 O.g.f.: Sum_{n>=0} n^n(2n+1)^n * exp(-n(2n+1)x) x^n / n!.