A106533 -id:A106533 - 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)).

A226750 Decimal expansion of the number x other than -3 defined by x*e^x = -3/e^3.

Original entry on oeis.org

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

Offset: 0

José María Grau Ribas, Jun 16 2013

			-0.178560627877921106596808669705514804654118255926885907720142306058785...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A106533, A217913, A221214, A274712, A274713.

RealDigits[N[ProductLog[-3/e^3], 105]][[1]]

solve(x=0,1,3/exp(3)-x*exp(-x)) \\ Charles R Greathouse IV, Nov 19 2013

Term a(104) corrected by G. C. Greubel, Nov 16 2017

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

Original entry on oeis.org

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

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

A226469 Decimal expansion of the maximum value reached by the function -2xlog(x)-2x(1-x) in the interval (0,1].

Original entry on oeis.org

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

Offset: 0

José María Grau Ribas, Jun 08 2013

Equals 2(c-c^2) where c = A106533 (the rumor constant).

			0.3238051189459574298236009809879772906045103360482538554...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A106533.

EE = -1/2 ProductLog[-2/(E^2)]; RealDigits[N[2*(EE-EE^2),100]][[1]]

{a = solve(x=0.1, x=0.5, 2*log(x) - 4*x + 4)}; -2*a*log(a)-2*a*(1-a) \\ G. C. Greubel, Nov 16 2017

Offset corrected by Rick L. Shepherd, Jan 02 2014

More terms from Alois P. Heinz, Jan 11 2014

A282025 a(r) is the maximum number of secretaries for which the first r should be rejected, if selecting the one with the highest or lowest ranking are both considered a success.

Original entry on oeis.org

3, 8, 13, 18, 23, 27, 32, 37, 42, 47, 52, 57, 62, 67, 72, 77, 82, 86, 91, 96, 101, 106, 111, 116, 121, 126, 131, 136, 141, 146, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 214, 219, 224, 229, 234, 239, 244, 249, 254, 259, 264, 269, 273, 278, 283

Offset: 0

N. J. A. Sloane, Feb 11 2017

nonn

According to Bayon et al, the probability P(n,r) = 2*r*((r/n-1)+sum_{i=r..n-1} 1/i)/n of success in a generalized Secretary problem for a given number n of applicants has a maximum at some value of r, 1<=r

The Beatty sequence of A106533, b(n) = floor(n*A106533), is a good approximation to r for large n. So the indices n-1 of the steps where b(n) = b(n+1)-1 are an approximation to this sequence.

We added numbers 27, 86 and 91 that are apparently missing in the preprint. R. J. Mathar, Feb 22 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..202
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.

Cf. A106533, A054404.

P := proc(n,
    option remember;
    local i;
    2*r/n*((r/n-1)+add(1/i,i=r..n-1)) ;
end proc:
Pmax := proc(n)
    option remember;
    local r;
    for r from 1 to n-1 do
        if P(n,r+1) < P(n,r) then
            return r ;
        end if;
    end do:
end proc:
A282025 := proc(r)
    local n ;
    if r = 0 then
        return 3;
    end if;
    for n from r+1 do
        if Pmax(n+1) = r+1 then
            return n;
        end if;
    end do:
    return -1 ;
end proc:
seq(A282025(r),r=0..80) ; # R. J. Mathar, Feb 22 2017

P[n_, r_] := 2 r ((r/n - 1) + Sum[ 1/i, {i, r, n - 1}])/n; Function[s, {3}~Join~Map[-1 + Position[s, #][[1, 1]] &, Range@ Max@ s]]@ Map[Length@ TakeWhile[#, # == 0 &] &, Table[If[P[n, k + 1] < P[n, k], k, 0], {n, 300}, {k, n - 1}]] (* Michael De Vlieger, Feb 22 2017, after Maple *)

A307951 Decimal expansion of 1 - log(2)/log(-W(-2/e^2)), where W is Lambert's W function.

Original entry on oeis.org

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

Offset: 1

Charles R Greathouse IV, May 07 2019

Chang shows that a constant population of n individuals, with ancestors selected uniformly at random, converges in probability to a state where every individual leaves either no current ancestors or else is a common ancestor of all present individuals after k*log_2(n) generations, where k is this constant (see Theorem 2 in Chang link for precise statement).

			1.769755495564801280059561457905786683522251513088978630155101689614415...
A population of 1000 is expected to have identical ancestors after around k*log_2(1000) = 17.6... generations.
A population of a million is expected to have identical ancestors after around k*log_2(10^6) = 35.2... generations.
A population of a billion is expected to have identical ancestors after around k*log_2(10^9) = 52.9... generations.
A population of a trillion is expected to have identical ancestors after around k*log_2(10^12) = 70.5... generations.

Joseph T. Chang, Recent common ancestors of all present-day individuals, Advances in Applied Probability Vol. 31, No. 4 (Dec., 1999), pp. 1002-1026.
James Grime and Brady Haran, EVERY baby is a ROYAL baby, Numberphile video (2019).
Douglas L. T. Rohde, Steve Olson, and Joseph T. Chang, Modelling the recent common ancestry of all living humans, Nature Vol. 431, No. 7008 (Sep. 2004), pp. 562-566.

Cf. A106533, A226775.

RealDigits[1 - Log[2]/Log[-ProductLog[-2/E^2]], 10, 120][[1]] (* Amiram Eldar, Jun 27 2023 *)

1 - log(2)/log(-lambertw(-2/exp(2))) \\ Charles R Greathouse IV, Jan 24 2025

Showing 1-9 of 9 results.

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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A226750 Decimal expansion of the number x other than -3 defined by x*e^x = -3/e^3.

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

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Maple

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Maxima

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

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

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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!.

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

A226469 Decimal expansion of the maximum value reached by the function -2xlog(x)-2x(1-x) in the interval (0,1].