A036244 -id:A036244 - cp's OEIS frontend

A058798 a(n) = n*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.

0, 1, 2, 5, 18, 85, 492, 3359, 26380, 234061, 2314230, 25222469, 300355398, 3879397705, 54011212472, 806288789375, 12846609417528, 217586071308601, 3903702674137290, 73952764737299909, 1475151592071860890

Offset: 0

Christian G. Bower, Dec 02 2000

Note that a(n) = (a(n-1) + a(n+1))/(n+1). - T. D. Noe, Oct 12 2012; corrected by Gary Detlefs, Oct 26 2018
a(n) = log_2(A073888(n)) = log_3(A073889(n)).
a(n) equals minus the determinant of M(n+2) where M(n) is the n X n symmetric tridiagonal matrix with entries 1 just above and below its diagonal and diagonal entries 0, 1, 2, .., n-1. Example: M(4)=matrix([[0, 1, 0, 0], [1, 1, 1, 0], [0, 1, 2, 1], [0, 0, 1, 3]]). - Roland Bacher, Jun 19 2001
a(n) = A221913(n,-1), n>=1, is the numerator sequence of the n-th approximation of the continued fraction -(0 + K_{k>=1} (-1/k)) = 1/(1-1/(2-1/(3-1/(4-... The corresponding denominator sequence is A058797(n). - Wolfdieter Lang, Mar 08 2013
The recurrence equation a(n+1) = (A*n + B)*a(n) + C*a(n-1) with the initial conditions a(0) = 0, a(1) = 1 has the solution a(n) = Sum_{k = 0..floor((n-1)/2)} C^k*binomial(n-k-1,k)*( Product_{j = 1..n-2k-1} (k+j)*A + B ). This is the case A = 1, B = 1, C = -1. - Peter Bala, Aug 01 2013

			Continued fraction approximation 1/(1-1/(2-1/(3-1/4))) = 18/7 = a(4)/A058797(4). - _Wolfdieter Lang_, Mar 08 2013

Seiichi Manyama, Table of n, a(n) for n = 0..449
Svante Janson, A divergent generating function that can be summed and analysed analytically, Discrete Mathematics and Theoretical Computer Science; 2010, Vol. 12, No. 2, 1-22.

Column 1 of A007754.
Cf. A073888, A073889, A221913 (alternating row sums).
Other recurrences of this type: A001040, A036242, A036244, A053983, A053984, A053987, A058307, A058308, A058309, A058797, A058799, A075374, A106174, A121323, A121351, A121353, A121354, A222468, A222470.
Similar sequences: A000806, A001053, A007754, A025164, A093986, A159927, A222467, A222469.

a:=[1,2];; for n in [3..25] do a[n]:=n*a[n-1]-a[n-2]; od; Concatenation([0], a); # Muniru A Asiru, Oct 26 2018

[0] cat [n le 2 select n else n*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 22 2016

t = {0, 1}; Do[AppendTo[t, n*t[[-1]] - t[[-2]]], {n, 2, 25}]; t (* T. D. Noe, Oct 12 2012 *)
nxt[{n_,a_,b_}]:={n+1,b,b*(n+1)-a}; Transpose[NestList[nxt,{1,0,1},20]] [[2]] (* Harvey P. Dale, Nov 30 2015 *)

m=30; v=concat([1,2], vector(m-2)); for(n=3, m, v[n] = n*v[n-1]-v[n-2]); concat(0, v) \\ G. C. Greubel, Nov 24 2018

def A058798(n):
    if n < 3: return n
    return hypergeometric([1/2-n/2, 1-n/2],[2, 1-n, -n], -4)*factorial(n)
[simplify(A058798(n)) for n in (0..20)] # Peter Luschny, Sep 10 2014

a(n) = Sum_{k = 0..floor((n-1)/2)} (-1)^k*binomial(n-k-1,k)*(n-k)!/(k+1)!. - Peter Bala, Aug 01 2013
a(n) = A058797(n+1) + A058799(n-1). - Henry Bottomley, Feb 28 2001
a(n) = Pi*(BesselY(1, 2)*BesselJ(n+1, 2) - BesselJ(1,2)* BesselY(n+1,2)). See the Abramowitz-Stegun reference given under A103921, p. 361 eq. 9.1.27 (first line with Y, J and z=2) and p. 360, eq. 9.1.16 (Wronskian). - Wolfdieter Lang, Mar 05 2013
Limit_{n->oo} a(n)/n! = BesselJ(1,2) = 0.576724807756873... See a comment on asymptotics under A084950.
a(n) = n!*hypergeometric([1/2-n/2, 1-n/2], [2, 1-n, -n], -4) for n >= 2. - Peter Luschny, Sep 10 2014

New description from Amarnath Murthy, Aug 17 2002

A053984 a(n) = (2n-1)a(n-1) - a(n-2), a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 3, 14, 95, 841, 9156, 118187, 1763649, 29863846, 565649425, 11848774079, 271956154392, 6787055085721, 182978531160075, 5299590348556454, 164104322274089999, 5410143044696413513, 189190902242100382956, 6994653239913017755859, 272602285454365592095545

Offset: 0

Vladeta Jovovic, Apr 02 2000

Numerators of successive convergents to tan(1) using continued fraction 1/(1-1/(3-1/(5-1/(7-1/(9-1/(11-1/(13-1/15-...))))))).

Equals eigensequence of an infinite lower triangular matrix with (1, 3, 5, 7, ...) as the main diagonal and (0, -1, -1, -1, ...) as the subdiagonal. - Gary W. Adamson, Apr 20 2009

			a(10)=565649425 because 1/(1-1/(3-1/(5-1/(7-1/(9-1/(11-1/(13-1/(15-1/(17-1/19))))))))) = 565649425/363199319.

G. C. Greubel, Table of n, a(n) for n = 0..400
S. Janson, A divergent generating function that can be summed and analysed analytically, Discrete Mathematics and Theoretical Computer Science; 2010, Vol. 12, No. 2, 1-22.

Cf. A053983, A058798.

[n le 2 select (n-1) else (2*n-3)*Self(n-1)-Self(n-2): n in [1..25] ]; // Vincenzo Librandi, May 12 2015

f:= gfun:-rectoproc({a(n)=(2*n-1)*a(n-1)-a(n-2),a(0)=0,a(1)=1},a(n),remember):
map(f, [$0..30]); # Robert Israel, May 14 2015

CoefficientList[Series[Sin[1-Sqrt[1-2*x]]/Sqrt[1-2*x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 05 2013 *)
RecurrenceTable[{a[n] == (2*n - 1)*a[n - 1] - a[n - 2], a[0] == 0,
  a[1] == 1}, a, {n, 0, 50}] (* G. C. Greubel, Jan 22 2017 *)

a(n)={if(n<2,n,(2*n-1)*a(n-1)-a(n-2))} \\ Edward Jiang, Sep 10 2014

{a(n) = my(a0, a1, s=n<0); if( abs(n) < 2, return(n)); if( n<0, n=-1-n); a0=s; a1=1; for(k=2, n, a2 = (2*k-1)*a1 - a0; a0=a1; a1=a2); (-1)^(s*n) * a1}; /* Michael Somos, Sep 11 2014 */

def A053984(n):
    if n < 2: return n
    return 2^n*gamma(n+1/2)*hypergeometric([1-n/2, 1/2-n/2],[3/2, 1 - n, 1/2 -n], -1) / sqrt(pi)
[round(A053984(n).n(100)) for n in (0..20)] # Peter Luschny, Sep 10 2014

a(n) = (-1)^n*A053983(-1-n). - Michael Somos, Aug 23 2000 [See Somos's formula in A053983 which is valid for all n in Z.]
E.g.f.: sin(1-sqrt(1-2*x))/sqrt(1-2*x). Cf. A036244. - Vladeta Jovovic, Aug 10 2006
Recurrence equation: a(n+1) = (2*n+1)*a(n) - a(n-1) with a(0) = 0 and a(1) = 1.
a(n) = Sum_{k = 0..floor((n-1)/2)} (-1)^k*2^(n-2*k-1)*(n-2*k-1)!*binomial(n-k-1,k)*binomial(n-k-1/2,k+1/2), cf. A058798. - Peter Bala, Aug 01 2013
a(n) ~ sin(1)*2^(n+1/2)*n^n/exp(n). - Vaclav Kotesovec, Oct 05 2013
a(n) = (2*n-1)!!*hypergeometric([1 - n/2, 1/2 - n/2], [3/2, 1 - n, 1/2 - n], -1) for n >= 2. - Peter Luschny, Sep 10 2014
0 = a(n)*(+a(n+2)) + a(n+1)*(-a(n+1) + 2*a(n+2) - a(n+3)) + a(n+2)*(+a(n+2)) for all n in Z. - Michael Somos, Sep 11 2014
a(n) = SphericalBesselJ[n,1]*SphericalBesselY[0,1] - SphericalBesselJ[0,1]*SphericalBesselY[n,1]. - G. C. Greubel, May 10 2015
Sum_{n>=0} a(n-1)*t^n/n! = - cos(1 - sqrt(1-2*t)), where a(-1) = -1. - G. C. Greubel, May 10 2015
The SphericalBessel formula given by Greubel above can be rewritten as a(n) = sqrt(Pi/2)*(-cos(1)*BesselJ(n+1/2, 1)  + (-1)^n*sin(1)*BesselJ(-(n+1/2), 1)). - Wolfdieter Lang, Jun 14 2015

Additional comments from Michael Somos, Aug 23 2000

More terms from Vladeta Jovovic, Aug 10 2006

A025164 a(n) = a(n-2) + (2n-1)a(n-1); a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 4, 21, 151, 1380, 15331, 200683, 3025576, 51635475, 984099601, 20717727096, 477491822809, 11958013297321, 323343850850476, 9388929687961125, 291380164177645351, 9624934347550257708, 337164082328436665131, 12484695980499706867555, 487240307321817004499776

Offset: 0

Wouter Meeussen

nonn

Numerators of convergents to coth(1) = 1.313035... = A073747.
Numerator of continued fraction given by C(n) = [ 1; 3, 5, 7, ..., (2n-1)]. - Amarnath Murthy, May 02 2001
Equals eigensequence of an infinite lower triangular matrix with (1, 3, 5, ...) in the main diagonal, (1, 1, 1, ...) in the sum diagonal, and the rest zeros. - Gary W. Adamson, Apr 17 2009
We can use the defining recurrence to extend the sequence to negative indices to give a(-n) = A036244(n-1). - Peter Bala, Sep 11 2014

			G.f. = 1 + x + 4*x^2 + 21*x^3 + 151*x^4 + 1380*x^5 + 15331*x^6 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..200
S. Janson, A divergent generating function that can be summed and analysed analytically, Discrete Mathematics and Theoretical Computer Science; 2010, Vol. 12, No. 2, 1-22.

Cf. A001040, A001053, A036244, A058798, A073747.

[n le 2 select 1 else (2*n-3)*Self(n-1)+Self(n-2): n in [1..20]]; // Vincenzo Librandi, Apr 22 2015

a:= proc(n) option remember;
      `if`(n<2, 1, a(n-2) +(2*n-1)*a(n-1))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 17 2014

a[ n_ ] := a[n] =a[n-2]+(-1+2 n) a[n-1]; a[0] := 1; a[1] := 1;
RecurrenceTable[{a[0]==a[1]==1,a[n]==a[n-2]+(2n-1)a[n-1]},a,{n,20}] (* Harvey P. Dale, Mar 25 2012 *)
a[ n_] := Round[ (Exp[1] + Exp[-1]) (BesselK[n - 3/2, 1] + (2 n - 1) BesselK[n - 1/2, 1]) / Sqrt[2 Pi]]; (* Michael Somos, Aug 26 2015 (n>=0) *)
a[ n_] := Module[{ y = Sqrt[1 - 2 x]}, n! SeriesCoefficient[ Cosh[y - 1] / y, {x, 0, n}]]; (* Michael Somos, Aug 26 2015 (n>=0) *)
a[ n_] := (BesselK[ 1/2, 1] BesselI[ n + 1/2, -1] - BesselI[ -1/2, -1] BesselK[ n + 1/2, 1]) / I // FunctionExpand // Simplify; (* Michael Somos, Aug 26 2015 *)
Join[{1}, Convergents[Coth[1], 20] // Numerator] (* Jean-François Alcover, Jun 15 2019 *)

a(n)={if(n<2,1,a(n-2)+(2*n-1)*a(n-1))} \\ Edward Jiang, Sep 11 2014

def A025164(n):
    if n == 0: return 1
    return sloane.A001147(n)*hypergeometric([-n/2+1/2, -n/2], [1/2, -n, 1/2-n], 1)
[round(A025164(n).n(100)) for n in (0..20)] # Peter Luschny, Sep 11 2014

E.g.f.: cosh((1-2*x)^(1/2)-1)/(1-2*x)^(1/2). - Vladeta Jovovic, Jan 30 2004
a(n) = round((exp(1)+exp(-1))*(BesselK(n-3/2, 1)+(2*n-1)*BesselK(n-1/2, 1))/sqrt(2*Pi) ). - Mark van Hoeij, Jul 02 2010
a(n) ~ sqrt(2)*cosh(1)*(2*n)^n/exp(n). - Vaclav Kotesovec, Jan 05 2013
a(n) = A001147(n)*hypergeometric([-n/2+1/2, -n/2], [1/2, -n, 1/2-n], 1) for n >= 1. - Peter Luschny, Sep 11 2014
a(n) = Sum_{k = 0..floor(n/2)} binomial(n - k, k)*( Product_{j = 1 .. n - 2*k} (2*k + 2*j - 1) ) = Sum_{k = 0..floor((n+1)/2)} 2^(2*k - n - 1)*(2*n + 2 - 2*k)!/( (n + 1 - 2*k)!*(2*k)! ). - Peter Bala, Sep 11 2014
a(n) = -i*( BesselK(1/2, 1)*BesselI(n+1/2, -1) - BesselI(-1/2, -1)*BesselK(n+1/2, 1)) for n>=0 (a(0)=1, a(1) = 1). - G. C. Greubel, Apr 21 2015
a(n) = A036244(-1-n) for all n in Z.
0 = a(n)*(-a(n+2)) + a(n+1)*(+a(n+1) + 2*a(n+2) - a(n+3)) + a(n+2)*(+a(n+2)) if n >= 0. - Michael Somos, Jan 10 2017
Given e.g.f. A(x), then 0 = A(x) + 3*A'(x) + (2*x-1)*A''(x). - Michael Somos, Jan 10 2017
Given g.f. A(x), then 0 = 1 + (x^2+x-1)*A(x) + 2*x^2*A'(x). - Michael Somos, Jan 10 2017

More terms from Larry Reeves (larryr(AT)acm.org), May 15 2001

More terms from Vladeta Jovovic, Jan 30 2004

cp's OEIS Frontend

A058798 a(n) = n*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

Extensions

A053984 a(n) = (2*n-1)*a(n-1) - a(n-2), a(0) = 0, a(1) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

Extensions

A025164 a(n) = a(n-2) + (2n-1)a(n-1); a(0)=1, a(1)=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A053984 a(n) = (2n-1)a(n-1) - a(n-2), a(0) = 0, a(1) = 1.