A217905 -id:A217905 - 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

A217901 O.g.f.: Sum_{n>=0} 2n^n (n+2)^(n-1) * exp(-n(n+2)x) * x^n / n!.

Original entry on oeis.org

1, 2, 10, 106, 1736, 38414, 1073178, 36281032, 1441336688, 65849949118, 3403003693310, 196336487214234, 12513043615743360, 873250527590532680, 66241197525447027832, 5427563864923583687376, 477771405475710621697632, 44970647131664087237328798, 4507506792104658670610331462

Offset: 0

Paul D. Hanna, Oct 14 2012

nonn

Compare the g.f. to the LambertW identity:

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

			O.g.f.: A(x) = 1 + 2*x + 10*x^2 + 106*x^3 + 1736*x^4 + 38414*x^5 + 1073178*x^6 +...
where
A(x) = 1 + 2*1^1*3^0*x*exp(-1*3*x) + 2*2^2*4^1*exp(-2*4*x)*x^2/2! + 2*3^3*5^2*exp(-3*5*x)*x^3/3! + 2*4^4*6^3*exp(-4*6*x)*x^4/4! + 2*5^5*7^4*exp(-5*7*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. A217899, A217900, A217902, A217903, A217904, A217905.

Flatten[{1,Table[Sum[Binomial[n-1,j]*2^(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,2*m^m*(m+2)^(m-1)*x^m*exp(-m*(m+2)*x+x*O(x^n))/m!),n)}

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

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

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

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

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

A256500 Decimal expansion of the positive solution to x = 2*(1-exp(-x)).

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Mar 31 2015

Each of the positive solutions to x = q*(1-exp(-x)) obtained for q = 2, 3, 4, and 5, appears in several formulas pertinent to Planck's black-body radiation law. For a given q, the solution can be also written as q+W(-q/exp(q)), where W is the Lambert function. Here q = 2.

The constant appears in asymptotic formula for A007820. - Vladimir Reshetnikov, Oct 10 2016

			1.5936242600400400923230418758751602417890024248188593649995...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 249.
SpectralCalc, Calculation of Blackbody Radiance, Appendix C.
Wikipedia, Planck's law
Index entries for transcendental numbers

Cf. A194567 (q=3), A256501 (q=4), A256502 (q=5).

Cf. A106533, A191236, A217905, A229553.

RealDigits[2 + LambertW[-2 Exp[-2]], 10, 100][[1]] (* Vladimir Reshetnikov, Oct 10 2016 *)

a2=solve(x=0.1,10,x-2*(1-exp(-x))) \\ Use real precision in excess

Equals 2*(1-A106533). - Miko Labalan, Dec 18 2024

Equals log(A229553). - Hugo Pfoertner, Dec 19 2024

A191236 Number of ways to place n nonattacking bishops on black squares of a 2n X 2n board.

Original entry on oeis.org

1, 2, 14, 184, 3532, 89256, 2800016, 104967808, 4578528464, 227816059360, 12735645181536, 790296855912576, 53905019035510528, 4008716449677965312, 322807879692969879552, 27983800239966141382656

Offset: 0

Vaclav Kotesovec, May 27 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 0..340
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 253.

Cf. A002465, A187235, A217905.

Join[{1}, Table[(1/n!)*Sum[(-1)^(n - k)*Binomial[n, k]*(k*(k + 1))^n, {k, 0, n}], {n,1,50}]] (* G. C. Greubel, Feb 03 2017 *)

{a(n)=polcoeff(sum(m=0,n,m^m*(m+1)^m*x^m*exp(-m*(m+1)*x+x*O(x^n))/m!),n)} \\ Paul D. Hanna, Oct 15 2012

{a(n)=sum(k=0,n, binomial(n,k) * stirling(2*n-k,n,2))} \\ Paul D. Hanna, Nov 13 2012

a(n) = 1/n! * Sum_{j=0..n} (-1)^(n-j) * binomial(n,j) * (j*(j+1))^n.
Asymptotic: a(n) ~ 1/sqrt(Pi*(z-1)*(2-z)*n)*(2*n*exp(z-1)/z)^n or a(n) ~ exp(z/2)*Stirling2(2*n,n) where z = A256500 = 1.59362426... is a root of the equation exp(z)*(2-z)=2.
O.g.f.: Sum_{n>=0} n^n*(n+1)^n * exp(-n*(n+1)*x) * x^n/n! = Sum_{n>=0} a(n)*x^n. - Paul D. Hanna, Oct 15 2012
a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(2*n-k,n), where Stirling2(n,k) = A008277(n,k). - Paul D. Hanna, Nov 13 2012

Offset changed to 0 and a(0)=1 added by Paul D. Hanna, Nov 13 2012

A217906 O.g.f.: 1 / Sum_{n>=0} -n^n(n-1)^(n-1) exp(-n(n-1)x) * x^n / n!.

Original entry on oeis.org

1, 1, 3, 19, 223, 4019, 98071, 3012595, 111408735, 4813926235, 237893755847, 13230156372931, 817650834368367, 55588558619887179, 4122802071853330711, 331247290236326404499, 28660436738240190615167, 2656810905539387715877787, 262694577305483845458361767

Offset: 0

Paul D. Hanna, Oct 14 2012

nonn

			O.g.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 223*x^4 + 4019*x^5 + 98071*x^6 +...
where
A(x) = 1/(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.

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A217905.

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

Convolution inverse of A217905.

a(n) ~ 2^(2*n - 2) * n^(n - 3/2) / (sqrt(Pi) * sqrt(1-c) * exp(n) * c^(n - 1/2) * (2-c)^(n-1)), where c = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599... - Vaclav Kotesovec, Aug 22 2018

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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A217901 O.g.f.: Sum_{n>=0} 2*n^n * (n+2)^(n-1) * exp(-n*(n+2)*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!.

A217901 O.g.f.: Sum_{n>=0} 2n^n (n+2)^(n-1) * exp(-n(n+2)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!.

A217906 O.g.f.: 1 / Sum_{n>=0} -n^n(n-1)^(n-1) exp(-n(n-1)x) * x^n / n!.