A256500 -id:A256500 - cp's OEIS frontend

A007820 Stirling numbers of second kind S(2n,n).

1, 1, 7, 90, 1701, 42525, 1323652, 49329280, 2141764053, 106175395755, 5917584964655, 366282500870286, 24930204590758260, 1850568574253550060, 148782988064375309400, 12879868072770626040000, 1194461517469807833782085, 118144018577011378596484455

Offset: 0

kemp(AT)sads.informatik.uni-frankfurt.de (Rainer Kemp)

Chan and Manna prove that a(n) is odd if and only if n is in A003714. - Jason Kimberley, Sep 14 2009
The number of ways to partition a set of 2*n elements into n disjoint subsets. - Vladimir Reshetnikov, Oct 10 2016
Conjecture: a(2*n+1) is divisible by (2*n + 1)^2. - Peter Bala, Mar 30 2025

			G.f.: A(x) = 1 + x + 7*x^2 + 90*x^3 + 1701*x^4 + 42525*x^5 +...,
where A(x) = 1 + 1^2*x*exp(-1*x) + 2^4*exp(-2^2*x)*x^2/2! + 3^6*exp(-3^2*x)*x^3/3! + 4^8*exp(-4^2*x)*x^4/4! + 5^10*exp(-5^2*x)*x^5/5! + ... - _Paul D. Hanna_, Oct 17 2012

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.

Alois P. Heinz, Table of n, a(n) for n = 0..345 (terms n = 1..200 from Vincenzo Librandi)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
O-Y. Chan and D. V. Manna, Divisibility properties of Stirling numbers of the second kind [From _Jason Kimberley_, Sep 14 2009]

Cf. A002465, A008277, A187646, A191236, A217913, A217914, A217915, A247238.

Cf. A350376, A383862, A384060.

A007820 := proc(n) Stirling2(2*n,n) ; end proc:
seq(A007820(n),n=0..20) ; # R. J. Mathar, Mar 15 2011

Table[StirlingS2[2n, n], {n, 1, 12}] (* Emanuele Munarini, Mar 12 2011 *)

makelist(stirling2(2*n,n),n,0,12); /* Emanuele Munarini, Mar 12 2011 */

a(n)=stirling(2*n,n,2); /* Joerg Arndt, Jul 01 2011 */

{a(n)=polcoeff(1/prod(k=1, n, 1-k*x +x*O(x^(2*n))), n)} \\ Paul D. Hanna, Oct 17 2012

{a(n)=polcoeff(sum(m=1,n,(m^2)^m*exp(-m^2*x+x*O(x^n))*x^m/m!),n)} \\ Paul D. Hanna, Oct 17 2012

from sympy.functions.combinatorial.numbers import stirling
def A007820(n): return stirling(n<<1,n) # Chai Wah Wu, Jun 09 2025

[stirling_number2(2*i,i) for i in range(1,20)] # Zerinvary Lajos, Jun 26 2008

a(n) = A048993(2n,n). - R. J. Mathar, Mar 15 2011
Asymptotic: a(n) ~ (4*n/(e*z*(2-z)))^n/sqrt(2*Pi*n*(z-1)), where z = A256500 = 1.59362426... is a root of the equation exp(z)*(2-z)=2. - Vaclav Kotesovec, May 30 2011
a(n) = 1/n! * Sum_{k = 0..n} binomial(n,k)*(-1)^k*(n-k)^(2*n). - Emanuele Munarini, Jul 01 2011
a(n) = [x^n] 1 / Product_{k=1..n} (1-k*x). - Paul D. Hanna, Oct 17 2012
O.g.f.: Sum_{n>=1} (n^2)^n * exp(-n^2*x) * x^n/n! = Sum_{n>=1} S2(2*n,n)*x^n. - Paul D. Hanna, Oct 17 2012
G.f.: Sum_{n > 0} (a(n)*n!/(2*n)!)*x^n = x*B'(x)/B(x)-1, where B(x) satisfies B(x)^2 = x*(exp(B(x))-1). - Vladimir Kruchinin, Mar 13 2013
a(n) = Sum_{j = 0..n} (-1)^(n-j)*n^j*binomial(2*n,j)*stirling2(2*n-j,n). - Vladimir Kruchinin, Jun 14 2013

Typo in Mathematica program fixed by Vincenzo Librandi, May 04 2013

a(0)=1 prepended by Alois P. Heinz, Feb 01 2018

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

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

A094090 Decimal expansion of positive solution to 5(1-exp(u)) + uexp(u) = 0.

Original entry on oeis.org

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

Offset: 1

Jeppe Stig Nielsen, May 01 2004

This purely mathematical constant turns up when in physics one derives Wien's displacement law from the Planck black-body radiation law (see link).

Positive solution to x = 5*(1-exp(-x)). More comments in A256500. - Stanislav Sykora, Apr 01 2015

			u=4.965114231744276...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
NIST, Wien displacement law constant, in Fundamental Physical Constants.
Eric Weisstein's World of Physics, Wien's Displacement Law
Wikipedia, Planck's law
Index entries for transcendental numbers

Cf. A194567, A256500, A256501.

RealDigits[5 + ProductLog[ -5/E^5], 10, 120][[1]] (* Robert G. Wilson v, May 04 2004 *)

a5=solve(x=0.1, 10, x-5*(1-exp(-x))) \\ Use real precision in excess

u = 5 + W(-5*exp(-5)), where W() is Lambert's W-function.

More terms from Robert G. Wilson v, May 04 2004

A194567 Decimal expansion of the positive solution to x = 3*(1-exp(-x)).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Aug 29 2011

The positive solution to x=3*(1-exp(-x)) is the dimensionless coefficient corresponding to the maximum brightness in Planck's law of radiation.

It can be symbolically expressed as 3+W(-3/e^3), where W stands for Lambert (a.k.a. "ProductLog") function.

			2.821439372...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
SpectralCalc, Calculation of Blackbody Radiance, Appendix C.
Wikipedia, Planck's law
Index entries for transcendental numbers

Cf. A094090, A256500, A256501.

evalf(3+LambertW(-3/exp(3)), 130);  # Alois P. Heinz, May 08 2024

RealDigits[ N[x /. ToRules[ Reduce[x > 0 && x == 3*(1 - E^-x), x, Reals]], 100]][[1]]
RealDigits[3 + ProductLog[-3/E^3], 10, 111][[1]] (* Robert G. Wilson v, Oct 16 2013 *)
RealDigits[x/.FindRoot[x==3(1-Exp[-x]),{x,2},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Aug 09 2023 *)

a3=solve(x=0.1,10,x-3*(1-exp(-x))) \\ Use real precision in excess

3+lambertw(-3/exp(3)) \\ Charles R Greathouse IV, Sep 13 2022

A256501 Decimal expansion of the positive solution to x = 4*(1-exp(-x)).

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Apr 01 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 = 4.

			3.9206903948728863435608913526135362205256273712079845304011750...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
SpectralCalc, Calculation of Blackbody Radiance, Appendix C.
Wikipedia, Planck's law
Index entries for transcendental numbers

Cf. A094090 (q=5), A194567 (q=3), A256500 (q=2).

RealDigits[x/.FindRoot[x==4(1-Exp[-x]),{x,3},WorkingPrecision->120]] [[1]] (* Harvey P. Dale, May 08 2017 *)

a4=solve(x=0.1, 10, x-4*(1-exp(-x))) \\ Use real precision in excess

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A007820 Stirling numbers of second kind S(2n,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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A191236 Number of ways to place n nonattacking bishops on black squares of a 2n X 2n board.

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A094090 Decimal expansion of positive solution to 5*(1-exp(u)) + u*exp(u) = 0.

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A194567 Decimal expansion of the positive solution to x = 3*(1-exp(-x)).

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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!.

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

A094090 Decimal expansion of positive solution to 5(1-exp(u)) + uexp(u) = 0.