A019762 -id:A019762 - cp's OEIS frontend

A062971 a(n) = (2*n)^n.

1, 2, 16, 216, 4096, 100000, 2985984, 105413504, 4294967296, 198359290368, 10240000000000, 584318301411328, 36520347436056576, 2481152873203736576, 182059119829942534144, 14348907000000000000000, 1208925819614629174706176, 108428035605965932354207744

Offset: 0

Jason Earls, Jul 23 2001

Shift n^n left n bits.

Also the number of input-closed output-Boolean Moore machines on n states. - David Spivak, Feb 14 2020

			n=3: 3^3 shifted three bits to the left is 216 because 3^3 in binary is: [1, 1, 0, 1, 1] and 216 in binary is: [1, 1, 0, 1, 1, 0, 0, 0].

Harry J. Smith, Table of n, a(n) for n = 0..100

Column k=1 of A246070.

Cf. A019762 (2*e).

a:= n-> (2*n)^n: seq(a(n), n=0..15); # Zerinvary Lajos, Jan 01 2009

Join[{1}, Table[(2*n)^n, {n,1,50}]] (* G. C. Greubel, Nov 10 2017 *)

for(n=0, 20, print1(shift(n^n,n), ", "))

E.g.f.: -(2*x*e^(-W(-2*x)))/(W(-2*x)*(W(-2*x)+1)), W(x) is Lambert's function. - Vladimir Kruchinin, May 09 2013
E.g.f.: 1/(1 + LambertW(-2*x)). - Vaclav Kotesovec, Dec 21 2014
Limit_{n->oo} a(n+1)/(n*a(n)) = 2*e. - Stefano Spezia, Mar 13 2023

New description from Vladeta Jovovic, Mar 08 2003

A052746 a(0) = 0; a(n) = (2*n)^(n-1), n > 0.

Original entry on oeis.org

0, 1, 4, 36, 512, 10000, 248832, 7529536, 268435456, 11019960576, 512000000000, 26559922791424, 1521681143169024, 95428956661682176, 6502111422497947648, 478296900000000000000, 37778931862957161709568, 3189059870763703892770816, 286511799958070431838109696

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Expansion of inverse of x*exp(2x).
Number of well-colored directed trees on n nodes. Well-colored means, each green vertex has at least a red child, each red vertex has no red child.
Number of labeled rooted directed trees on n nodes.

Robert Israel, Table of n, a(n) for n = 0..350
Federico Ardila, Matthias Beck, and Jodi McWhirter, The Arithmetic of Coxeter Permutahedra, arXiv:2004.02952 [math.CO], 2020.
C. Banderier, J.-M. Le Bars, and V. Ravelomanana, Generating functions for kernels of digraphs, arXiv:math/0411138 [math.CO], 2004.
Theo Douvropoulos, Joel Brewster Lewis, and Alejandro H. Morales, Hurwitz numbers for reflection groups III: Uniform formulas, arXiv:2308.04751 [math.CO], 2023, see p. 11.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 702

Cf. A019762 (2*e), A038057, A097627.

spec := [S,{B=Set(S),S=Prod(Z,B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

terms = 19;
A[x_] = -1/2 LambertW[-2 x];
CoefficientList[A[x] + O[x]^terms, x] Range[0, terms-1]! (* Jean-François Alcover, Jan 15 2019 *)
Join[{0},Table[(2n)^(n-1),{n,20}]] (* Harvey P. Dale, Dec 14 2020 *)

a(n)=if(n,(2*n)^(n-1),0) \\ Charles R Greathouse IV, Nov 20 2011

[lucas_number1(n, 2*n, 0) for n in range(0, 17)] # Zerinvary Lajos, Mar 09 2009

E.g.f.: -1/2*W(-2*x), where W is Lambert's W function.
From Robert Israel, Jun 16 2016: (Start)
E.g.f. g(x) satisfies g(x) = x*exp(2*g(x)) and (1-2*g(x)) g'(x) = g(x).
a(n) = (2*n/(n-1)) * Sum_{j=1..n-1} binomial(n-1,j)*a(j)*a(n-j) for n >= 2. (End)
a(n) = [x^n] x/(1 - 2*n*x). - Ilya Gutkovskiy, Oct 12 2017
Limit_{n->oo} a(n+1)/(n*a(n)) = 2*e. - Stefano Spezia, Mar 12 2023

New description from Vladeta Jovovic, Mar 08 2003

A019463 Add 1, multiply by 1, add 2, multiply by 2, etc., start with 1.

Original entry on oeis.org

1, 2, 2, 4, 8, 11, 33, 37, 148, 153, 765, 771, 4626, 4633, 32431, 32439, 259512, 259521, 2335689, 2335699, 23356990, 23357001, 256927011, 256927023, 3083124276, 3083124289, 40080615757, 40080615771, 561128620794, 561128620809, 8416929312135, 8416929312151, 134670868994416

Offset: 0

N. J. A. Sloane

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..900
Nick Hobson, Python program for this sequence.

Cf. A019461 (same, but start with 0), A019460 (start with 2), A019462, (start with 3), A082448. (start with 4).
Cf. A082458, A019464, A019465, A019466 (similar, but first multiply, then add).
Cf. A019762.

a:= proc(n) option remember; `if`(n=0, 1, (t->
      `if`(n::odd, t+(n+1)/2, t*n/2))(a(n-1)))
    end:
seq(a(n), n=0..32);  # Alois P. Heinz, Jan 16 2024

For[i=1;lst={1},i<15,i++,AppendTo[lst,i+Last[lst]];AppendTo[lst,i Last[lst]]];lst (* Harvey P. Dale, Feb 25 2012 *)
FoldList[If[OddQ[#2], #1 + (#2 + 1)/2, #1 * (#2/2)]&, 1, Range[32]] (* AnneMarie Torresen, Nov 26 2023 *)

A019463(n, a=1)={for(i=2, n+1, if(bittest(i, 0), a*=i\2, a+=i\2)); a} \\ M. F. Hasler, Feb 25 2018

Limit_{n->oo} a(2n)/n! = 1 + 2e = 1 + A019762. - Jon E. Schoenfield, Jan 16 2024

Edited by M. F. Hasler, Feb 25 2018

A038057 a(n) = 2^n*n^(n-1).

Original entry on oeis.org

2, 8, 72, 1024, 20000, 497664, 15059072, 536870912, 22039921152, 1024000000000, 53119845582848, 3043362286338048, 190857913323364352, 13004222844995895296, 956593800000000000000, 75557863725914323419136

Offset: 1

Christian G. Bower, Jan 04 1999

nonn

Labeled rooted trees with n 2-colored nodes.

Index entries for sequences related to rooted trees

Cf. A000169, A019762 (2*e), A038055-A038062.

Equals 2 * A052746(n).

nn=16;f[x_]:=Sum[a[n]x^n/n!,{n,0,nn}];s=SolveAlways[0==Series[f[x]-2x Exp[f[x]],{x,0,nn}],x];Table[a[n],{n,1,nn}]/.s
(* or *)
nn=16;Drop[Range[0,nn]!CoefficientList[Series[-LambertW[-2x],{x,0,nn}],x],1]
(* or *)
Table[2^n*n^(n-1),{n,1,16}]  (* Geoffrey Critzer, Mar 17 2013 *)

E.g.f.: B(2*x) where B(x) is e.g.f. of A000169.

Limit_{n->oo} a(n+1)/(n*a(n)) = 2*e. - Stefano Spezia, Mar 12 2023

New description from Vladeta Jovovic, Mar 08 2003

A081750 Simple continued fraction of 2*e.

Original entry on oeis.org

5, 2, 3, 2, 3, 1, 2, 1, 3, 4, 3, 1, 4, 1, 3, 6, 3, 1, 6, 1, 3, 8, 3, 1, 8, 1, 3, 10, 3, 1, 10, 1, 3, 12, 3, 1, 12, 1, 3, 14, 3, 1, 14, 1, 3, 16, 3, 1, 16, 1, 3, 18, 3, 1, 18, 1, 3, 20, 3, 1, 20, 1, 3, 22, 3, 1, 22, 1, 3, 24, 3, 1, 24, 1, 3, 26, 3, 1, 26, 1, 3, 28, 3, 1, 28, 1, 3, 30, 3, 1, 30, 1, 3, 32

Offset: 1

Benoit Cloitre, Apr 08 2003

Decimal expansion is A019762. - Michael Somos, May 07 2012

			2*e = 5 + 1 / (2 + 1 / (3 + 1 / (2 + 1 / (3 + 1 / (1 + ...))))). - _Michael Somos_, May 07 2012

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,1,0,0,-1).

Cf. A006083, A006084, A006085, A019762.

ContinuedFraction[ 2E, 94] (* Robert G. Wilson v, May 07 2012 *)

A081750(n) = if(1==n,5,if(n<6,2+(n%2), my(k=n\6, r=n%6); if(0==r || 2==r, 1, if(1==r, 2*k, if(n%2, 3, 2*(k+1)))))); \\ Antti Karttunen, Feb 20 2023

First 5 terms are 5, 2, 3, 2, 3.

For k >= 1, a(6k)=1; a(6k+1)=2k; a(6k+2)=1; a(6k+3)=3; a(6k+4)=2k+2; a(6k+5)=3.

A230404 a(n) = the largest k such that (k+1)! divides 2n; the number of trailing zeros in the factorial base representation of even numbers.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1

Offset: 1

Antti Karttunen, Oct 31 2013

Antti Karttunen, Table of n, a(n) for n = 1..10080

Used to compute A230405 and A219650. See A007623 for factorial base representation.
Analogous sequence for binary system: A001511.
Cf. A019762.

Scheme
```
(define (A230404 n) (A230403 (* 2 n)))
```

a(n) = A230403(2n) = A055881(2n) - 1.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*e - 4 = A019762 - 4 = 1.436563... . - Amiram Eldar, Jan 05 2024

A081749 Continued fraction for e/5.

Original entry on oeis.org

0, 1, 1, 5, 4, 2, 2, 2, 2, 2, 1, 1, 9, 1, 1, 3, 3, 2, 3, 3, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 3, 9, 1, 3, 3, 3, 4, 3, 3, 4, 1, 2, 2, 1, 4, 1, 2, 2, 1, 5, 9, 1, 5, 3, 3, 6, 3, 3, 6, 1, 2, 2, 1, 6, 1, 2, 2, 1, 7, 9, 1, 7, 3, 3, 8, 3, 3, 8, 1, 2, 2, 1, 8, 1, 2, 2, 1, 9, 9, 1, 9, 3, 3, 10, 3, 3, 10, 1, 2, 2, 1, 10, 1, 2

Offset: 1

Benoit Cloitre, Apr 08 2003

Ray Chandler, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1).

Cf. A019762 (decimal expansion).

Cf. A003417 (e), A006083 (e/2), A006084 (e/3), A006085 (e/4).

ContinuedFraction[E/5, 100] (* Paolo Xausa, Sep 21 2024 *)

contfrac(exp(1)/5) \\ Michel Marcus, Dec 03 2013

First 18 terms: 0, 1, 1, 5, 4, 2, 2, 2, 2, 2, 1, 1, 9, 1, 1, 3, 3, 2.

For k >= 1, a(19k)=a(19k+1)=a(19k+16)=a(19k+17)=3; a(19k+2)=a(19k+7)=2k; a(19k+3)=a(19k+6)=a(19k+8)=a(19k+11)=a(19k+14)=1; a(19k+4)=a(19k+5)=a(19k+9)= a(19k+10)=2; a(19k+12)=a(19k+15)=2k+1; a(19k+18)=2k+2.

A361291 a(n) = ((2n + 1)^n - 1)/(2n).

Original entry on oeis.org

1, 6, 57, 820, 16105, 402234, 12204241, 435984840, 17927094321, 833994048910, 43309534450633, 2483526865641276, 155867505885345241, 10627079738421409410, 782175399728156197665, 61812037545704964935440, 5220088150634922700769761, 469168161404536131943150998

Offset: 1

Stefano Spezia, Mar 12 2023

This sequence is of the form (k^n - 1)/(k - 1) with k = 2*n + 1. See crossrefs in A218722 for other sequences of the same form.

Cf. A005408, A019762 (2*e), A051129, A218722.

Cf. A000169, A000312, A038057, A052746, A062971, A213236.

Mathematica
```
Table[((2n+1)^n-1)/(2n),{n,20}]
```

def A361291(n): return (((n<<1)+1)**n-1)//(n<<1) # Chai Wah Wu, Mar 14 2023

a(n) = Sum_{i=0..n-1} A005408(n)^i.
a(n) = n! * [x^n] exp(x)*(exp(2*n*x) - 1)/(2*n).
a(n) = n! * [x^n] exp((n+1)*x)*sinh(n*x)/n.
Limit_{n->oo} a(n+1)/(n*a(n)) = 2*e.
Limit_{n->oo} (a(n+1)/a(n) - a(n)/a(n-1)) = 2*e.

A365307 Decimal expansion of 1/(2*e-5).

Original entry on oeis.org

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

Offset: 1

Rok Cestnik, Aug 31 2023

The continued fraction expansion is A081750 with initial term 5 omitted.

			2.2906166927853624221...

Cf. A001113, A194807, A019762, A113011, A081750, A073333, A185108.

A365307 = RealDigits[N[1/(2*E-5),#+1]][[1]][[1;;-2]]&;

PARI
```
1/(2*exp(1)-5).
```

Equals 2 + 1/(3 + 2/(4 + 3/(5 + 4/(6 + 5/( ... /(n+1 + n/(n+2 + ... ))))))).
From Peter Bala, Oct 23 2023: (Start)
Define s(n) = Sum_{k = 3..n} 1/k!. Then 1/(2*e - 5) = 3 - (1/2)*Sum_{n >= 3 } 1/( (n+1)!*s(n)*s(n+1) ) is a rapidly converging series of rationals. Cf. A073333 and A194807.
Equivalently, 1/(2*e - 5) = 3 - (1/2)*(3!/(1*5) + 4!/(5*26) + 5!/(26*157) + 6!/(157*1100) + ...), where [1, 5, 26, 157, 1100, ... ] is A185108. (End)

A248859 Decimal expansion of log(sqrt(2*Pi))/e, a constant appearing in the asymptotic expansion of (n!)^(1/n).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Mar 03 2015

			0.3380585940662399023702794509615188742685137583402...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2021, p. 57.
Shafiqur Rahman and Leonard Giugiuc, Problem 4285, Crux Mathematicorum, Vol. 43, No. 9 (2017), pp. 399 and 401; Solution to Problem 4285, ibid., Vol. 44, No. 9 (2018), p. 395.

Cf. A001113, A019762, A061444, A075700 (log(sqrt(2*Pi))).

SetDefaultRealField(RealField(100)); R:= RealField(); Log(2*Pi(R))/(2*Exp(1)); // G. C. Greubel, Oct 07 2018

RealDigits[Log[Sqrt[2*Pi]]/E, 10, 105] // First

log(2*Pi)/2/exp(1) \\ Charles R Greathouse IV, Apr 20 2016

Equals lim_{n -> infinity} (n!)^(1/n) - n/e - log(n)/(2*e).

Equals A075700/A001113 = A061444/A019762. - Amiram Eldar, Apr 12 2022

cp's OEIS Frontend

A062971 a(n) = (2*n)^n.

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A052746 a(0) = 0; a(n) = (2*n)^(n-1), n > 0.

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A019463 Add 1, multiply by 1, add 2, multiply by 2, etc., start with 1.

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A038057 a(n) = 2^n*n^(n-1).

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A081750 Simple continued fraction of 2*e.

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A230404 a(n) = the largest k such that (k+1)! divides 2n; the number of trailing zeros in the factorial base representation of even numbers.

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A081749 Continued fraction for e/5.

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A361291 a(n) = ((2n + 1)^n - 1)/(2n).