A113012 -id:A113012 - cp's OEIS frontend

A000354 Expansion of e.g.f. exp(-x)/(1-2*x).

1, 1, 5, 29, 233, 2329, 27949, 391285, 6260561, 112690097, 2253801941, 49583642701, 1190007424825, 30940193045449, 866325405272573, 25989762158177189, 831672389061670049, 28276861228096781665, 1017967004211484139941, 38682746160036397317757

Offset: 0

N. J. A. Sloane

a(n) is the permanent of the n X n matrix with 1's on the diagonal and 2's elsewhere. - Yuval Dekel, Nov 01 2003. Compare A157142.
Starting with offset 1 = lim_{k->infinity} M^k, where M = a tridiagonal matrix with (1,0,0,0,...) in the main diagonal, (1,3,5,7,...) in the subdiagonal and (2,4,6,8,...) in the subsubdiagonal. - Gary W. Adamson, Jan 13 2009
a(n) is also the number of (n-1)-dimensional facet derangements for the n-dimensional hypercube. - Elizabeth McMahon, Gary Gordon (mcmahone(AT)lafayette.edu), Jun 29 2009
a(n) is the number of ways to write down each n-permutation and underline some (possibly none or all) of the elements that are not fixed points. a(n) = Sum_{k=0..n} A008290(n,k)*2^(n-k). - Geoffrey Critzer, Dec 15 2012
Type B derangement numbers: the number of fixed point free permutations in the n-th hyperoctahedral group of signed permutations of {1,2,...,n}. See Chow 2006. See A000166 for type A derangement numbers. - Peter Bala, Jan 30 2015

			G.f. = 1 + x + 5*x^2 + 29*x^3 + 233*x^4 + 2329*x^5 + 27949*x^6 + 391285*x^7 + ... - _Michael Somos_, Apr 14 2018

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 83.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Chak-On Chow, On derangement polynomials of type B, Séminaire Lotharingien de Combinatoire 55 (2006), Article B55b.
Gary Gordon and Elizabeth McMahon, Moving faces to other places: Facet derangements, arXiv:0906.4253 [math.CO], 2009.
Gary Gordon and Elizabeth McMahon, Moving faces to other places: facet derangements, Amer. Math. Monthly, 117 (2010), 865-88.
Édouard Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 223.
Édouard Lucas, Théorie des nombres (annotated scans of a few selected pages)
István Mezo, Victor H. Moll, José L. Ramírez, and Diego Villamizar, On the r-Derangements of type B, arXiv:2103.04151 [math.CO], 2021.
István Mező, Victor H. Moll, José Ramírez, and Diego Villamizar, On the r-derangements of type B, Online Journal of Analytic Combinatorics, Issue 16 (2021), #05.
Jean-Christophe Pain, A sum rule for r-derangements obtained from the Cauchy product of exponential generating functions, arXiv:2408.15927 [math.CO], 2024.
Simon Plouffe, Exact formulas for integer sequences
L. W. Shapiro & N. J. A. Sloane, Correspondence, 1976
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Cf. A061714, A008290, A000166, A113012, A263895.

Column k=2 of A320032.

a := n -> (-1)^n*(1-2*n*hypergeom([1,1-n],[],2)):
seq(simplify(a(n)), n=0..18); # Peter Luschny, May 09 2017
a := n -> 2^n*add((n!/k!)*(-1/2)^k, k=0..n):
seq(a(n), n=0..23); # Peter Luschny, Jan 06 2020
seq(simplify(2^n*KummerU(-n, -n, -1/2)), n = 0..19); # Peter Luschny, May 10 2022

FunctionExpand @ Table[ Gamma[ n+1, -1/2 ]*2^n/Exp[ 1/2 ], {n, 0, 24}]
With[{nn=20},CoefficientList[Series[Exp[-x]/(1-2x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 22 2013 *)
a[n_] := 2^n n! Sum[(-1)^i/(2^i i!), {i, 0, n}]; Table[a[n], {n, 0, 20}] (* Gerry Martens, May 06 2016 *)
a[ n_] := If[ n < 1, Boole[n == 0], (2 n - 1) a[n - 1] + (2 n - 2) a[n - 2]]; (* Michael Somos, Sep 28 2017 *)
a[ n_] := Sum[ (-1)^(n + k) Binomial[n, k] k! 2^k, {k, 0, n}]; (* Michael Somos, Apr 14 2018 *)
a[ n_] := If[ n < 0, 0, (2^n Gamma[n + 1, -1/2]) / Sqrt[E] // FunctionExpand]; (* Michael Somos, Apr 14 2018 *)
a[n_] := n! 2^n Hypergeometric1F1[-n, -n, -1/2];
Table[a[n], {n, 0, 19}]   (* Peter Luschny, Jul 28 2024 *)

my(x='x+O('x^66)); Vec(serlaplace(exp(-x)/(1-2*x))) \\ Joerg Arndt, Apr 15 2013

vector(100, n, n--; sum(k=0, n, (-1)^(n+k)*binomial(n, k)*k!*2^k)) \\ Altug Alkan, Oct 30 2015

{a(n) = if( n<1, n==0, (2*n - 1) * a(n-1) + (2*n - 2) * a(n-2))}; /* Michael Somos, Sep 28 2017 */

Inverse binomial transform of double factorials A000165. - Paul Barry, May 26 2003
a(n) = Sum_{k=0..n} (-1)^(n+k)*C(n, k)*k!*2^k. - Paul Barry, May 26 2003
a(n) = Sum_{k=0..n} A008290(n, k)*2^(n-k). - Philippe Deléham, Dec 13 2003
a(n) = 2*n*a(n-1) + (-1)^n, n > 0, a(0)=1. - Paul Barry, Aug 26 2004
D-finite with recurrence a(n) = (2*n-1)*a(n-1) + (2*n-2)*a(n-2). - Elizabeth McMahon, Gary Gordon (mcmahone(AT)lafayette.edu), Jun 29 2009
From Groux Roland, Jan 17 2011: (Start)
a(n) = (1/(2*sqrt(exp(1))))*Integral_{x>=-1} exp(-x/2)*x^n dx;
Sum_{k>=0} 1/(k!*2^(k+1)*(n+k+1)) = (-1)^n*(a(n)*sqrt(exp(1))-2^n*n!). (End)
a(n) = round(2^n*n!/exp(1/2)), x >= 0. - Simon Plouffe, Mar 1993
G.f.: 1/Q(0), where Q(k) = 1 - x*(4*k+1) - 4*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 30 2013
From Peter Bala, Jan 30 2015: (Start)
a(n) = Integral_{x = 0..inf} (2*x - 1)^n*exp(-x) dx.
b(n) := 2^n*n! satisfies the recurrence b(n) = (2*n - 1)*b(n-1) + (2*n - 2)*b(n-2), the same recurrence as satisfied by a(n). This leads to the continued fraction representation a(n) = 2^n*n!*( 1/(1 + 1/(1 + 2/(3 + 4/(5 +...+ (2*n - 2)/(2*n - 1) ))))) for n >= 2, which in the limit gives the continued fraction representation sqrt(e) = 1 + 1/(1 + 2/(3 + 4/(5 + ... ))). (End)
For n > 0, a(n) = 1 + 4*Sum_{k=0..n-1} A263895(n). - Vladimir Reshetnikov, Oct 30 2015
a(n) = (-1)^n*(1-2*n*hypergeom([1,1-n],[],2)). - Peter Luschny, May 09 2017
a(n+1) >= A113012(n). - Michael Somos, Sep 28 2017
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (2*k - 1) * a(n-k). - Ilya Gutkovskiy, Jan 17 2020
a(n) = 2^n*KummerU(-n, -n, -1/2). - Peter Luschny, May 10 2022
a(n) = 2^n*n!*hypergeom([-n], [-n], -1/2). - Peter Luschny, Jul 28 2024

A113011 Decimal expansion of 1/(sqrt(e) - 1).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, following a suggestion of Grover W. Hughes, Oct 09 2005

Has continued fraction 1+2/(3+4/(5+6/7+...)).

Simple continued fraction is 1, 1, 1, 5, 1, 1, 9, 1, 1, 13, 1, 1, 17, 1, {1, 4k+1, 1}, ..., . - Robert G. Wilson v, Jul 01 2007

			1.54149408253679828413110344447251463834045923684188210947413695663...

G. C. Greubel, Table of n, a(n) for n = 1..20000
Itay Beit-Halachmi and Ido Kaminer, The Ramanujan Library -- Automated Discovery on the Hypergraph of Integer Relations, arXiv:2412.12361 [cs.AI], 2024. See p. 6.
Leonhard Euler, On the formation of continued fractions, arXiv:math/0508227 [math.HO], 2005, see p. 14.
Michel Waldschmidt, Continued fractions, Ecole de recherche CIMPA-Oujda, Théorie des Nombres et ses Applications, 18 - 29 mai 2015: Oujda (Maroc).
Eric Weisstein's World of Mathematics, Continued Fraction.
Index entries for transcendental numbers.

Cf. A019774, A113012, A113013.

1/(Sqrt(Exp(1)) - 1); // G. C. Greubel, Apr 09 2018

First@ RealDigits[ 1 / (Exp[1/2] - 1), 10, 111] (* Robert G. Wilson v, Jul 01 2007 *)
f[n_] := Fold[ Last@ #2 + First@ #2/#1 &, 2n - 1, Partition[ Reverse@ Range[ 2n - 2], 2]]; RealDigits[ f[61], 10, 105][[1]] (* Robert G. Wilson v, Jul 07 2012 *)
Rest[realDigitsRecip[Sqrt[E]-1]] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Nov 04 2024 *)

1/(sqrt(exp(1)) - 1) \\ G. C. Greubel, Apr 09 2018

Equals Integral_{x = 0..oo} floor(2*x)*exp(-x) dx. - Peter Bala, Oct 09 2013
Equals 3/2 + Sum_{k>=0} tanh(1/2^(k+3))/2^(k+2). - Antonio Graciá Llorente, Jan 21 2024
Conjecture: 1/(sqrt(e) - 1) = 1 + K_{n>=1} 2*n/(4*n^2-1)/1, where K is the Gauss notation for an infinite continued fraction. In the expanded form, 1/(sqrt(e) - 1) = 1 + 2/3/(1 + 4/15/(1 + 6/35/(1 + ...))) (see Beit-Halachmi and Kaminer). - Stefano Spezia, Dec 27 2024
Equals 1/(A019774 - 1). - Hugo Pfoertner, Dec 27 2024

Simpler definition from T. D. Noe, Oct 09 2005

Euler reference from David L. Harden, Oct 09 2005

A113013 Denominators of convergents to 1 + 2/(3 + 4/(5 + 6/(7 + ...))).

Original entry on oeis.org

1, 3, 19, 151, 1511, 18131, 50767, 4061359, 73104463, 1462089259, 32165963699, 30879325151, 20071561348151, 562003717748227, 16860111532446811, 539523569038297951, 3668760269460426067, 660376848502876692059, 25094320243109314298243, 1003772809724372571929719, 42158458008423648021048199

Offset: 0

Eric W. Weisstein, Oct 10 2005

			1, 5/3, 29/19, 233/151, 2329/1511, ...

G. C. Greubel, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Eric Weisstein's World of Mathematics, Generalized Continued Fraction

Cf. A113011, A113012.

f[n_] := Fold[ Last@ #2 + First@ #2/#1 &, 2n - 1, Partition[ Reverse@ Range[ 2n - 2], 2]]; Denominator[ Array[ f, 18]] (* Robert G. Wilson v, Jul 07 2012 *)
(* It is interesting to note that FoldList[2 #1*#2 - (-1)^#2 &, 0, Range[19]] matches many of the terms. - Robert G. Wilson v, Jul 07 2012 *)
a[ n_] := If[ n < 0, 0, Denominator[ 1 + ContinuedFractionK[2 i, 2 i + 1, {i, 1, n}]]]; (* Michael Somos, Apr 14 2018 *)
Table[1 + ContinuedFractionK[2 k, 2 k + 1, {k, n}], {n, 0, 20}] // Denominator (* Eric W. Weisstein, Apr 14 2018 *)
Table[1/((Sqrt[E] Gamma[n + 2])/Gamma[n + 2, -1/2] - 1), {n, 0, 20}] // Denominator (* Eric W. Weisstein, Apr 14 2018 *)

{a(n) = my(A); if( n<0, 0, A = contfracpnqn( matrix(2, n, j, i, [2*i, 2*i+1] [j]) ); denominator( 1 + A[2, 1] / A[1, 1]) )}; /* Michael Somos, Apr 14 2018 */

a(n) = denominators of 1/((sqrt(e) * Gamma(n+2))/Gamma(n+2, -1/2) - 1), where Gamma(x, a) is the incomplete Gamma function. - Eric W. Weisstein, Apr 14 2018

A354298 a(n) is the numerator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.

Original entry on oeis.org

1, 2, 11, 76, 137, 7534, 97943, 1469144, 24975449, 94906706, 9965204131, 229199695012, 5729992375301, 9100576125478, 897316805972131, 563093542209232, 4589775462547450033, 5539384178936577626, 5943759223998947792699, 46361321947191792783052, 9504070999174317520525661

Offset: 1

Ilya Gutkovskiy, May 23 2022

			1, 2/3, 11/15, 76/105, 137/189, 7534/10395, 97943/135135, 1469144/2027025, 24975449/34459425, ...

Robert Israel, Table of n, a(n) for n = 1..403

Cf. A001147, A053557, A061354, A064646, A103816, A113012, A120265, A143382, A289381, A306858, A354299 (denominators).

S:= 0: R:= NULL:
for n from 1 to 100 do
  S:= S + (-1)^(n+1)/doublefactorial(2*n-1);
  R:= R, numer(S);
od:
R; # Robert Israel, Jan 10 2024

Table[Sum[(-1)^(k + 1)/(2 k - 1)!!, {k, 1, n}], {n, 1, 21}] // Numerator
nmax = 21; CoefficientList[Series[Sqrt[Pi x Exp[-x]/2] Erfi[Sqrt[x/2]]/(1 - x), {x, 0, nmax}], x] // Numerator // Rest
Table[1/(1 + ContinuedFractionK[2 k - 1, 2 k, {k, 1, n - 1}]), {n, 1, 21}] // Numerator

Numerators of coefficients in expansion of sqrt(Pi*x*exp(-x)/2) * erfi(sqrt(x/2)) / (1 - x).

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A000354 Expansion of e.g.f. exp(-x)/(1-2*x).

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A113011 Decimal expansion of 1/(sqrt(e) - 1).

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A113013 Denominators of convergents to 1 + 2/(3 + 4/(5 + 6/(7 + ...))).

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A354298 a(n) is the numerator of Sum_{k=1..n} (-1)^(k+1) / (2*k-1)!!.

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