A217900 -id:A217900 - cp's OEIS frontend

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

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

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

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

A218668 O.g.f.: Sum_{n>=0} 1/(1-n^2x)^n x^n/n! * exp(-x/(1-n^2*x)).

Original entry on oeis.org

1, 0, 1, 3, 22, 161, 1546, 18857, 270320, 4471693, 85455574, 1865128265, 45735737037, 1247518965519, 37654095184226, 1250673144714138, 45415758777730668, 1792734161930717221, 76595370803745016626, 3529261203030717032927, 174742139545017029583279

Offset: 0

Paul D. Hanna, Nov 04 2012

nonn

Compare g.f. to the curious identity:

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

			O.g.f.: A(x) = 1 + x^2 + 3*x^3 + 22*x^4 + 161*x^5 + 1546*x^6 + 18857*x^7 +...
where
A(x) = exp(-x) + x/(1-x)*exp(-x/(1-x)) + x^2/(1-4*x)^2/2!*exp(-x/(1-4*x)) + x^3/(1-9*x)^3/3!*exp(-x/(1-9*x)) + x^4/(1-16*x)^4/4!*exp(-x/(1-16*x)) + x^5/(1-25*x)^5/5!*exp(-x/(1-25*x)) +...
simplifies to a power series in x with integer coefficients.

Vaclav Kotesovec, Table of n, a(n) for n = 0..290

Cf. A218667, A218669, A218670, A217900.

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

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

A218669 O.g.f.: Sum_{n>=0} 1/(1-n^3x)^n x^n/n! * exp(-x/(1-n^3*x)).

Original entry on oeis.org

1, 0, 1, 7, 97, 1561, 41136, 1551814, 72440460, 4281320257, 324623105584, 30086950057627, 3299720918091511, 428431079916572044, 65637957066642609845, 11659659637028895337265, 2367270866164121777222596, 546795407830461739380895161, 143176487805296033192642234802

Offset: 0

Paul D. Hanna, Nov 04 2012

nonn

Compare g.f. to the curious identity:

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

			O.g.f.: A(x) = 1 + x^2 + 7*x^3 + 97*x^4 + 1561*x^5 + 41136*x^6 +...
where
A(x) = exp(-x) + x/(1-x)*exp(-x/(1-x)) + x^2/(1-8*x)^2/2!*exp(-x/(1-8*x)) + x^3/(1-27*x)^3/3!*exp(-x/(1-27*x)) + x^4/(1-64*x)^4/4!*exp(-x/(1-64*x)) + x^5/(1-125*x)^5/5!*exp(-x/(1-125*x)) +...
simplifies to a power series in x with integer coefficients.

Cf. A218667, A218668, A218670, A217900.

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

cp's OEIS Frontend

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

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

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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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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!.

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

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!.

A218668 O.g.f.: Sum_{n>=0} 1/(1-n^2x)^n x^n/n! * exp(-x/(1-n^2*x)).

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!.

A218669 O.g.f.: Sum_{n>=0} 1/(1-n^3x)^n x^n/n! * exp(-x/(1-n^3*x)).