A053518 -id:A053518 - cp's OEIS frontend

A053556 Denominator of Sum_{k=0..n} (-1)^k/k!.

1, 1, 2, 3, 8, 30, 144, 280, 5760, 45360, 44800, 3991680, 43545600, 172972800, 6706022400, 93405312000, 42268262400, 22230464256000, 376610217984000, 250298560512000, 11640679464960000, 196503623737344000, 17841281393295360000

Offset: 0

N. J. A. Sloane, Jan 17 2000

Denominator of probability of a derangement of n things (A000166(n)/n!).

Also numerators of successive convergents to e using continued fraction 2 +1/(1 +1/(2 +2/(3 +3/(4 +4/(5 +5/(6 +6/(7 +7/8 +...))))))).

			1, 0, 1/2, 1/3, 3/8, 11/30, 53/144, 103/280, 2119/5760, ...

L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 562.
E. Maor, e: The Story of a Number, Princeton Univ. Press 1994, pp. 151 and 157.

G. C. Greubel, Table of n, a(n) for n = 0..450 (terms 0..100 from T. D. Noe)
Leonhardo Eulero, Introductio in analysin infinitorum. Tomus primus, Lausanne, 1748.
L. Euler, Introduction à l'analyse infinitésimale, Tome premier, Tome second, trad. du latin en français par J. B. Labey, Paris, 1796-1797.
Eric Weisstein's World of Mathematics, Subfactorial

Cf. A053557 (numerators), A053518-A053520. See also A103816.

a(n) = (D(n, n) of A103360), A053557/A053556 = A000166/n! = (N(n, n) of A103361)/(D(n, n) of A103360).

[Denominator( (&+[(-1)^k/Factorial(k): k in [0..n]]) ): n in [0..20]]; // G. C. Greubel, May 16 2019

Table[Denominator[Sum[(-1)^k/k!, {k, 0, n}]], {n, 0, 20}] (* Robert G. Wilson v, Oct 13 2005 *)
Table[ Denominator[1 - Subfactorial[n]/n!], {n, 0, 22}] (* Jean-François Alcover, Feb 11 2014 *)
Denominator[Accumulate[Table[(-1)^k/k!,{k,0,30}]]] (* Harvey P. Dale, Aug 22 2016 *)

for(n=0,50, print1(denominator(sum(k=0,n,(-1)^k/k!)), ", ")) \\ G. C. Greubel, Nov 05 2017

from math import factorial
from fractions import Fraction
def A053556(n): return sum(Fraction(-1 if k&1 else 1,factorial(k)) for k in range(n+1)).denominator # Chai Wah Wu, Jul 31 2023

[denominator(sum((-1)^k/factorial(k) for k in (0..n))) for n in (0..20)] # G. C. Greubel, May 16 2019

Let exp(-x)/(1-x) = Sum_{n>=0} (a_n/b_n) * x^n. Then sequence b_n is A053556. - Aleksandar Petojevic, Apr 14 2004

More terms from Vladeta Jovovic, Mar 31 2000

A053557 Numerator of Sum_{k=0..n} (-1)^k/k!.

Original entry on oeis.org

1, 0, 1, 1, 3, 11, 53, 103, 2119, 16687, 16481, 1468457, 16019531, 63633137, 2467007773, 34361893981, 15549624751, 8178130767479, 138547156531409, 92079694567171, 4282366656425369, 72289643288657479, 6563440628747948887, 39299278806015611311

Offset: 0

N. J. A. Sloane, Jan 17 2000

Numerator of probability of a derangement of n things (A000166(n)/n! or !n/n!).

Also denominators of successive convergents to e using continued fraction 2 + 1/(1 + 1/(2 + 2/(3 + 3/(4 + 4/(5 + 5/(6 + 6/(7 + 7/(8 + ...)))))))).

			1, 0, 1/2, 1/3, 3/8, 11/30, 53/144, 103/280, 2119/5760, ...

L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 562.
E. Maor, e: The Story of a Number, Princeton Univ. Press 1994, pp. 151 and 157.

G. C. Greubel, Table of n, a(n) for n = 0..450 (terms 0..100 from T. D. Noe)
Leonhardo Eulero, Introductio in analysin infinitorum. Tomus primus, Lausanne, 1748.
L. Euler, Introduction à l'analyse infinitésimale, Tome premier, Tome second, trad. du latin en français par J. B. Labey, Paris, 1796-1797.
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Eric Weisstein's World of Mathematics, Generalized Continued Fraction
Eric Weisstein's World of Mathematics, Subfactorial

Cf. A000166/A000142, A053556 (denominators), A053518-A053520. See also A103816.

a(n) = (N(n, n) of A103361), A053557/A053556 = A000166/n! = (N(n, n) of A103361)/(D(n, n) of A103360), Cf. A053518-A053520.

[Numerator( (&+[(-1)^k/Factorial(k): k in [0..n]]) ): n in [0..30]]; // G. C. Greubel, May 16 2019

Numerator[CoefficientList[Series[Exp[-x]/(1-x), {x, 0, 30}], x]] (* Jean-François Alcover, Nov 18 2011 *)
Table[Numerator[Sum[(-1)^k/k!,{k,0,n}]],{n,0,30}] (* Harvey P. Dale, Dec 02 2011 *)
Join[{1, 0}, Numerator[RecurrenceTable[{a[n]==a[n-1]+a[n-2]/(n-2), a[1] ==0, a[2]==1}, a, {n,2,30}]]] (* Terry D. Grant, May 07 2017; corrected by G. C. Greubel, May 16 2019 *)

for(n=0, 30, print1(numerator(sum(k=0,n, (-1)^k/k!)), ", ")) \\ G. C. Greubel, Nov 05 2017

from fractions import Fraction
from math import factorial
def A053557(n): return sum(Fraction(-1 if k&1 else 1,factorial(k)) for k in range(n+1)).numerator # Chai Wah Wu, Jul 31 2023

[numerator(sum((-1)^k/factorial(k) for k in (0..n))) for n in (0..30)] # G. C. Greubel, May 16 2019

Let exp(-x)/(1-x) = Sum_{n >= 0} (a_n/b_n)*x^n. Then sequence a_n is A053557. - Aleksandar Petojevic, Apr 14 2004

More terms from Vladeta Jovovic, Mar 31 2000

A053520 Denominators of successive convergents to continued fraction 1/(2+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/9+...))))))).

Original entry on oeis.org

2, 8, 38, 74, 1522, 11986, 11838, 1054766, 11506538, 45706526, 1772006854, 24681524038, 11169012898, 5874202721042, 99515904921182, 66139171377658, 3075946152109262, 51924337160029042, 4714400135799462226

Offset: 0

N. J. A. Sloane, Jan 15 2000

			Convergents are 1/2, 3/8, 15/38, 29/74, 597/1522, 4701/11986, ...

M. A. Stern, Theorie der Kettenbrüche und ihre Anwendung, Crelle, 1832, pp. 1-22.

Cf. A053518, A053519, A053556, A053557.

for j from 1 to 50 do printf(`%d,`,denom(cfrac([0,seq([i,i+1],i=1..j)]))); od:

a[n_] := ContinuedFractionK[k, k+1, {k, n+1}] // Denominator; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Mar 21 2016 *)

More terms from James Sellers, Feb 02 2000

A053519 Denominators of successive convergents to continued fraction 1+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/(9+9/10+...))))))).

Original entry on oeis.org

1, 3, 15, 29, 597, 4701, 4643, 413691, 4512993, 17926611, 695000919, 9680369943, 4380611853, 2303928046437, 39031251610227, 25940523189513, 1206420504316107, 20365306128628437, 1849040492948486661

Offset: 0

N. J. A. Sloane, Jan 15 2000

Also numerators of successive convergents to continued fraction 1/(2+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/9+...))))))).

A053518/A053519 -> (2*e-5)/(3-e) = 1.5496467783... as n-> infinity.

			Convergents (to the first continued fraction) are 1, 5/3, 23/15, 45/29, 925/597, 7285/4701, ...

L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 562.
E. Maor, e: The Story of a Number, Princeton Univ. Press 1994, pp. 151 and 157.

Leonhardo Eulero, Introductio in analysin infinitorum. Tomus primus, Lausanne, 1748.
L. Euler, Introduction à l'analyse infinitésimale, Tome premier, Tome second, trad. du latin en français par J. B. Labey, Paris, 1796-1797.
M. A. Stern, Theorie der Kettenbrüche und ihre Anwendung, Crelle, 1832, pp. 1-22.

Cf. A053518, A053520, A053556, A053557.

for j from 1 to 50 do printf(`%d,`,denom(cfrac([1,seq([i,i+1],i=2..j)]))); od:

num[0]=1; num[1]=5; num[n_] := num[n] = (n+2)*num[n-1] + (n+1)*num[n-2]; den[0]=1; den[1]=3; den[n_] := den[n] = (n+2)*den[n-1] + (n+1)*den[n-2]; a[n_] := Denominator[num[n]/den[n]]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jan 16 2013 *)

Thanks to R. K. Guy, Steven Finch, Bill Gosper for comments

More terms from James Sellers, Feb 02 2000

A174458 Partial sums of A053519.

Original entry on oeis.org

1, 4, 19, 48, 645, 5346, 9989, 423680, 4936673, 22863284, 717864203, 10398234146, 14778845999, 2318706892436, 41349958502663, 67290481692176, 1273710986008283, 21639017114636720, 1870679510063123381

Offset: 0

Jonathan Vos Post, Mar 20 2010

Partial sums of denominators of successive convergents to continued fraction 1+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/(9+9/10+...))))))). The subsequence of primes in this partial sum begins: 19, 1273710986008283.

			a(16) = 1 + 3 + 15 + 29 + 597 + 4701 + 4643 + 413691 + 4512993 + 17926611 + 695000919 + 9680369943 + 4380611853 + 2303928046437 + 39031251610227 + 25940523189513 + 1206420504316107 = 1273710986008283 is prime.

Cf. A053519, A053518, A053520, A053556, A053557.

a(n) = SUM[i=0..n] A053519(i).

cp's OEIS Frontend

A053556 Denominator of Sum_{k=0..n} (-1)^k/k!.

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A053557 Numerator of Sum_{k=0..n} (-1)^k/k!.

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A053520 Denominators of successive convergents to continued fraction 1/(2+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/9+...))))))).

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A053519 Denominators of successive convergents to continued fraction 1+2/(3+3/(4+4/(5+5/(6+6/(7+7/(8+8/(9+9/10+...))))))).

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A174458 Partial sums of A053519.

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