A217899 -id:A217899 - 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

A226775 Decimal expansion of the number x other than -2 defined by x*exp(x) = -2/e^2.

Original entry on oeis.org

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

Offset: 0

José María Grau Ribas, Jun 17 2013

			-0.4063757399599599076769581241248397582109975751811406350004954883....

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers.

Cf. A106533, A187655, A187657, A217899, A217900, A226750, A243227, A245109, A247238, A258467, A337458.

RealDigits[N[ProductLog[-2/E^2], 105]][[1]] (* corrected by Vaclav Kotesovec, Feb 21 2014 *)

solve(x=-1, x=0, x*exp(x) + 2*exp(-2)) \\ G. C. Greubel, Nov 15 2017

Equals -2*A106533.

Equals LambertW(-2*exp(-2)).

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

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

A245109 G.f.: Sum_{n>=0} exp(-(1 + n^2x)) (1 + n^2*x)^n / n!.

Original entry on oeis.org

1, 3, 31, 520, 11991, 350889, 12428746, 516450792, 24619176153, 1323971052261, 79280864647205, 5231080689880500, 377062508515478306, 29479066783583059530, 2484534527715953700780, 224559818606249783480400, 21666961097367611148157815, 2222844864226101120054773295

Offset: 0

Paul D. Hanna, Jul 12 2014

nonn

Compare the g.f. to:
(1) Sum_{n>=0} exp(-(1+n*x)) * (1+n*x)^n / n! = 1/(1-x).
(2) Sum_{n>=1} exp(-n^2*x) * n^(2*n) * x^n/n! = Sum_{n>=1} S2(2*n,n)*x^n (A007820).

			G.f.: A(x) = 1 + 3*x + 31*x^2 + 520*x^3 + 11991*x^4 + 350889*x^5 +...
where
A(x) = exp(-1) + exp(-(1+x))*(1+x) + exp(-(1+2^2*x))*(1+2^2*x)^2/2!
+ exp(-(1+3^2*x))*(1+3^2*x)^3/3! + exp(-(1+4^2*x))*(1+4^2*x)^4/4!
+ exp(-(1+5^2*x))*(1+5^2*x)^5/5! + exp(-(1+6^2*x))*(1+6^2*x)^6/6!
+ exp(-(1+7^2*x))*(1+7^2*x)^7/7! + exp(-(1+8^2*x))*(1+8^2*x)^8/8! +...
simplifies to a power series in x with integer coefficients.

Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 0..200 (first 100 terms from Paul D. Hanna)

Cf. A049020, A245110.

Cf. A187655, A217899, A217900.

Table[SeriesCoefficient[Sum[E^(-(1+k^2*x))*(1+k^2*x)^k/k!,{k,0,Infinity}],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 12 2014 *)

/* Must first set suitable precision */ \p300
{a(n)=local(A=1+x); A=suminf(k=0, exp(-(1+k^2*x)+x*O(x^n))*(1+k^2*x)^k/k!); round(polcoeff(A, n))}
for(n=0, 30, print1(a(n), ", "))

a(n) ~ c * d^n * (n-1)!, where d = -4/(LambertW(-2*exp(-2))*(2+LambertW(-2*exp(-2)))) = 6.17655460948348035823168..., and c = 10.427337127699040838035... . - Vaclav Kotesovec, Jul 12 2014

a(n) = A049020(2n,n). - Alois P. Heinz, Aug 23 2017

A247238 a(n) = Stirling2(2*n+1, n).

Original entry on oeis.org

1, 15, 301, 7770, 246730, 9321312, 408741333, 20415995028, 1144614626805, 71187132291275, 4864251308951100, 362262620784874680, 29206898819153109600, 2534474684137526739000, 235535731151727520125765, 23339590705557273894321960

Offset: 1

Vladimir Kruchinin, Nov 28 2014

nonn

			O.g.f.: A(x) = x + 15*x^2 + 301*x^3 + 7770*x^4 + 246730*x^5 + 9321312*x^6 + ... where A(x) = 1^3*x*exp(-1^2*x) + 2^5*exp(-2^2*x)*x^2/2! + 3^7*exp(-3^2*x)*x^3/3! + 4^9*exp(-4^2*x)*x^4/4! + 5^11*exp(-5^2*x)*x^5/5! + ...

G. C. Greubel, Table of n, a(n) for n = 1..340

Cf. A048993, A007820, A217899, A243227, A226775.

Table[StirlingS2[2*n+1, n], {n, 1, 20}] (* Vaclav Kotesovec, Nov 29 2014 *)

vector(50, n, stirling(2*n+1, n, 2)) \\ Colin Barker, Nov 28 2014

a(n) = A243227(n) / (n-1)!. - Vaclav Kotesovec, Nov 29 2014
a(n) ~ 2^(2*n+1/2) * n^(n+1/2) / (sqrt(Pi) * sqrt(1-c) * exp(n) * c^n * (2-c)^(n+1)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... (see A226775). - Vaclav Kotesovec, Nov 29 2014
O.g.f. Sum_{n>=1} n^(2*n+1) * x^n * exp(-n^2*x) / n! = Sum_{n>=1} a(n)*x^n. - Paul D. Hanna, Oct 09 2023

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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A226775 Decimal expansion of the number x other than -2 defined by x*exp(x) = -2/e^2.

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

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

A245109 G.f.: Sum_{n>=0} exp(-(1 + n^2x)) (1 + n^2*x)^n / n!.