A209289 -id:A209289 - cp's OEIS frontend

A085984 Decimal expansion of solution to e^x*(-1 + x) = (1 + x)/e^x.

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

Offset: 1

Eric W. Weisstein, Jul 06 2003

This constant can also be defined as the root of coth x = x, as this equation and the above are equivalent. - Carl R. White, Dec 09 2003. Also the root of x*tanh x = 1. - N. J. A. Sloane, May 07 2020
This constant is also the point on the parametric tractrix (t - tanh(t), sech(t)) the least distant from the origin. - Michael Clausen, Feb 18 2013
This constant also equals sqrt(lambda^2+1), where lambda is the Laplace limit constant A033259. - Jean-François Alcover, Sep 08 2014, after Steven Finch.
For each of the real symmetric n X n matrices M defined by M(i,j) = max(i,j) with n >= 2, there exist n-1 negative eigenvalues < -1/4 and only one positive eigenvalue lambda(n) such that n^2/2 < lambda(n) < n^2. Indeed, when n tends to infinity, lambda(n) ~ n^2/(this constant)^2 (see reference O. Carton et al.). For n = 2, the positive eigenvalue is (3+sqrt(17))/2 [A178255]. - Bernard Schott, Mar 13 2020

			1.1996786402577338339163698486411419442614587884186072...

O. Carton, L. Rosaz, M. Zeitoun, Problèmes corrigés de Mathématiques posés au Concours de Mines/Ponts, Tome 5, Ellipses, 1992; Problème Mines-Ponts 1991 - Options M, P', TA - Epreuve pratique p. 125.
Steven R. Finch, Mathematical constants, Volume 94, Encyclopedia of mathematics and its applications, Cambridge University Press, 2003, p. 268.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 30, equation 30:10:9 at page 283.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Jogundas Armaitis, Molecules and Polarons in Extremely Imbalanced Fermi Mixtures, Master's Thesis, Aug 11 2011, Institute for Theoretical Physics, Utrecht University.
Robert Ferréol, Tractrix, Mathcurve.
D. E. Knuth, Whirlpool Permutations, May 05 2020.
Mathematics Stack Exchange, Laplace limit constant.
Eric Weisstein's World of Mathematics, Kepler's Equation.
Eric Weisstein's World of Mathematics, Laplace Limit.
Eric Weisstein's World of Mathematics, Hyperbolic Cotangent
Wikipedia, Tractrix.
Index entries for transcendental numbers.

Cf. A003957 (x = cos(x)), A009379, A033259, A069855, A209289.

RealDigits[ x /. FindRoot[ Coth[x] == x, {x, 1}, WorkingPrecision -> 102]] // First (* Jean-François Alcover, Feb 08 2013 *)
1+2 NSum[LaguerreL[n-1,1,4 n]/n Exp[-2 n],{n,1,Infinity}] //
  (* Aaron Hendrickson, Mar 17 2021 *)

solve(u=1,2,tanh(u)-1/u)  /* type e.g. \p99 to get 99 digits; M. F. Hasler, Feb 01 2011 */

Equals 1 + 2*Sum_{n>=1} (Laguerre(n-1,1,4n)/n)*e^(-2n) (see Mathematics Stack Exchange in Links). - Aaron Hendrickson, Mar 17 2022

A306800 Square array whose entry A(n,k) is the number of endofunctions on a set of size n with preimage constraint {0,1,...,k}, for n >= 0, k >= 0, read by descending antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 4, 6, 0, 1, 1, 4, 24, 24, 0, 1, 1, 4, 27, 204, 120, 0, 1, 1, 4, 27, 252, 2220, 720, 0, 1, 1, 4, 27, 256, 3020, 29520, 5040, 0, 1, 1, 4, 27, 256, 3120, 44220, 463680, 40320, 0, 1, 1, 4, 27, 256, 3125, 46470, 765030, 8401680, 362880, 0

Offset: 0

Benjamin Otto, Mar 10 2019

A preimage constraint is a set of nonnegative integers such that the size of the inverse image of any element is one of the values in that set.

Thus, A(n,k) is the number of endofunctions on a set of size n such that each preimage has at most k entries. Equivalently, A(n,k) is the number of n-letter words from an n-letter alphabet such that no letter appears more than k times.

			Array begins:
  1    1     1     1     1 ...
  0    1     1     1     1 ...
  0    2     4     4     4 ...
  0    6    24    27    27 ...
  0   24   204   252   256 ...
  0  120  2220  3020  3120 ...
  0  720 29520 44220 46470 ...
  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
B. Otto, Coalescence under Preimage Constraints, arXiv:1903.00542 [math.CO], 2019, Corollaries 5.6 and 7.8.

Columns 0-9: A000007, A000142, A012244, A239368, A324804, A324805, A324806, A324807, A324808, A324809.
A(n,n) gives A000312.
Similar array for preimage condition {i>=0 | i!=k}: A245413.
Number of functions with preimage condition given by the even nonnegative integers: A209289.
Sum over all k of the number of functions with preimage condition {0,k}: A231812.
Cf. A019575.

b:= proc(n, i, k) option remember; `if`(n=0 and i=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1, k)*binomial(n, j), j=0..min(k, n))))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Apr 05 2019

b[n_, i_, k_] := b[n, i, k] = If[n==0 && i==0, 1, If[i<1, 0, Sum[b[n-j, i-1, k] Binomial[n, j], {j, 0, Min[k, n]}]]];
A[n_, k_] := b[n, n, k];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 29 2019, after Alois P. Heinz *)

# print first num_entries entries in column k
import math, sympy; x=sympy.symbols('x')
k=5; num_entries = 64
P=range(k+1); eP=sum([x**d/math.factorial(d) for d in P]); r = [1]; curr_pow = 1
for term in range(1,num_entries):
   curr_pow=(curr_pow*eP).expand()
   r.append(curr_pow.coeff(x**term)*math.factorial(term))
print(r)

A(n,k) = n! * [x^n] e_k(x)^n, where e_k(x) is the truncated exponential 1 + x + x^2/2! + ... + x^k/k!. When k>1, the link above yields explicit constants c_k, r_k so that the columns are asymptotically c_k * n^(-1/2) * r_k^-n. Stirling's approximation gives column k=1, and column k=0 is 0.

A(n,k) = Sum_{j=1..min(k,n)} A019575(n,j) for n>=1. - Alois P. Heinz, Jun 28 2023

Offset changed to 0 by Alois P. Heinz, Jun 28 2023

A383853 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(4*n).

Original entry on oeis.org

1, 1, 260, 556032, 4641176128, 106519579045760, 5472276566891956224, 549375993583284180705280, 97867116732573493470161420288, 28783909470167571938915053763592192, 13216052972619446942074113385580542689280, 9058922175695195359062480694771506779050213376

Offset: 0

Vaclav Kotesovec, May 12 2025

nonn

Cf. A209289, A298851, A345876.

Join[{1}, Table[Sum[Binomial[2*n, k]*(n-k)^(4*n), {k, 0, n}], {n, 1, 12}]] (* or *)
Join[{1}, Table[Sum[Binomial[2*n, n+k]*k^(4*n), {k, 0, n}], {n, 1, 12}]]

a(n) = Sum_{k=0..n} binomial(2*n, n+k) * k^(4*n).

a(n) ~ 4^n * r^(4*n+1) * n^(4*n) / (sqrt(2 - r^2) * (1 - r^2)^n * exp(4*n)), where r = 0.9683644349844134852843167967986294187258222293516... is the root of the equation (1+r)/(1-r) = exp(4/r).

A383916 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(3*n).

Original entry on oeis.org

1, 1, 68, 22770, 21143488, 41904629550, 151957171590144, 910666718387157732, 8390164064875701321728, 112583179357513548960803670, 2109812207969377622615440752640, 53397692462483465346961668429307836, 1775866125092261344436828225211633500160, 75857512919848315654302238627976991244564300

Offset: 0

Vaclav Kotesovec, May 15 2025

nonn

Cf. A032443, A345876, A209289/2, A383853, A383917.

Join[{1}, Table[Sum[Binomial[2*n, n-k]*k^(3*n), {k, 0, n}], {n, 1, 15}]]

a(n) ~ 2^(2*n + 1/2) * r^(3*n + 1) * n^(3*n) / (sqrt(3 - r^2) * exp(3*n) * (1 - r^2)^n), where r = 0.92488761106894648930384927930334708844525256369797556858640... is the root of the equation (1 + r)/(1 - r) = exp(3/r).

A383917 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(5*n).

Original entry on oeis.org

1, 1, 1028, 14545530, 1127435263168, 309320354959336350, 232325928732003715014144, 403150958104730561230009068564, 1432706082674749593552098155989352448, 9528431104471630510834164178027409070527670, 110580781643902847320855308323644986008860441968640

Offset: 0

Vaclav Kotesovec, May 15 2025

nonn

In general, for m>=1, Sum_{k=0..n} binomial(2*n, n-k) * k^(m*n) ~ 2^(2*n + 1/2) * r^(m*n + 1) * n^(m*n) / (sqrt(m + (2-m)*r^2) * exp(m*n) * (1 - r^2)^n), where r is the root of the equation (1 + r)/(1 - r) = exp(m/r).

Cf. A032443 (m=0), A345876 (m=1), A209289/2 (m=2), A383916 (m=3), A383853 (m=4).

Join[{1}, Table[Sum[Binomial[2*n, n-k]*k^(5*n), {k, 0, n}], {n, 1, 12}]]

a(n) ~ 2^(2*n + 1/2) * r^(5*n + 1) * n^(5*n) / (sqrt(5 - 3*r^2) * exp(5*n) * (1 - r^2)^n), where r = 0.98743428968604456152277643726278132237092161504496484119319... is the root of the equation (1 + r)/(1 - r) = exp(5/r).

A225942 Triangular array read by rows: T(n,k) is the number of f:{1,2,...,n}->{1,2,...,n} with exactly 2k elements that have a preimage of even (possibly zero) cardinality; n>=0, 0<=k<=floor(n/2).

Original entry on oeis.org

1, 1, 2, 2, 6, 21, 24, 192, 40, 120, 1800, 1205, 720, 18000, 25680, 2256, 5040, 194040, 489510, 134953, 40320, 2257920, 9031680, 5196800, 250496, 362880, 28304640, 167015520, 166793760, 24943689, 3628800, 381024000, 3149798400, 4904524800, 1514960640, 46063360

Offset: 0

Geoffrey Critzer, May 21 2013

Urn A is initially filled with n labeled balls while urn B is empty. A ball is randomly selected and switched from one urn to the other. T(n,k)/n^n is the probability that urn A contains 2k balls after n switches have been made.
Row sums = n^n.
T(n,0)   = n!.
T(2n,n)  = A209289(n).

			1;
1;
2,    2;
6,    21;
24,   192,    40;
120,  1800,   1205;
720,  18000,  25680,  2256;
5040, 194040, 489510, 134953;

Alois P. Heinz, Rows n = 0..100, flattened

Map[Select[#, # > 0 &] &, Prepend[Table[nn = n;
    CoefficientList[
     Expand[n! Coefficient[
        Series[(y Cosh[x] + Sinh[x])^n, {x, 0, nn}], x^n]], y], {n, 1,
      7}], {1}]] // Grid

T(n,k) = n! * [x^n*y^(2k)] (y*cosh(x)+sinh(x))^n.

cp's OEIS Frontend

A085984 Decimal expansion of solution to e^x*(-1 + x) = (1 + x)/e^x.

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A306800 Square array whose entry A(n,k) is the number of endofunctions on a set of size n with preimage constraint {0,1,...,k}, for n >= 0, k >= 0, read by descending antidiagonals.

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A383853 a(n) = Sum_{k=0..n} binomial(2*n, k) * (n-k)^(4*n).

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A383916 a(n) = Sum_{k=0..n} binomial(2*n, k) * (n-k)^(3*n).

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A383917 a(n) = Sum_{k=0..n} binomial(2*n, k) * (n-k)^(5*n).

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A225942 Triangular array read by rows: T(n,k) is the number of f:{1,2,...,n}->{1,2,...,n} with exactly 2k elements that have a preimage of even (possibly zero) cardinality; n>=0, 0<=k<=floor(n/2).

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Mathematica

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A383853 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(4*n).

A383916 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(3*n).

A383917 a(n) = Sum_{k=0..n} binomial(2n, k) (n-k)^(5*n).