A036242 -id:A036242 - 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

A036243 Denominator of fraction equal to the continued fraction [ 0, 2, 4, ...2n ].

Original entry on oeis.org

1, 2, 9, 56, 457, 4626, 55969, 788192, 12667041, 228794930, 4588565641, 101177239032, 2432842302409, 63355077101666, 1776375001149057, 53354605111573376, 1709123738571497089, 58163561716542474402, 2095597345534100575561

Offset: 0

Jeff Burch

Cf. A036242 (numerator), A369585.

b := n -> BesselK(n,1)*BesselI(0,1)-(-1)^n*BesselI(n,1)* BesselK(0,1);
A036243 := n -> b(n+1):
seq(simplify(A036243(n)), n=0..18); # Peter Luschny, Sep 14 2014

Table[Denominator[FromContinuedFraction[Range[0,2n,2]]],{n,0,20}] (* Harvey P. Dale, Feb 18 2016 *)

a(n)=contfracpnqn(vector(n+1,i,2*i-2))[2,1];
vector(22,n,a(n-1)) \\ M. F. Hasler, Feb 08 2011; edited by Michel Marcus, Feb 12 2024

a(n) = b(n+1) where b(n) = K(n,1)*I(0,1) - (-1)^n*I(n,1)*K(0,1), K(n,x) and I(n,x) Bessel functions. - Peter Luschny, Sep 14 2014
a(n) = Sum_{0..n} |A369585(n)|. - Peter Luschny, Jan 30 2024
a(n) = 2*n*a(n-1) + a(n-2). - Christian Krause, Aug 18 2024

a(0) = 1 prepended by Peter Luschny, Jan 30 2024

A246658 Triangle read by rows: T(n,k) = K(n,1)I(k,1) - (-1)^(n+k)I(n,1)* K(k,1), where I(n,x) and K(n,x) are Bessel functions; 0<=k<=n.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 9, 4, 1, 0, 56, 25, 6, 1, 0, 457, 204, 49, 8, 1, 0, 4626, 2065, 496, 81, 10, 1, 0, 55969, 24984, 6001, 980, 121, 12, 1, 0, 788192, 351841, 84510, 13801, 1704, 169, 14, 1, 0, 12667041, 5654440, 1358161, 221796, 27385, 2716, 225, 16, 1, 0

Offset: 0

Peter Luschny, Sep 14 2014

			     0;
     1,      0;
     2,      1,     0;
     9,      4,     1,     0;
    56,     25,     6,     1,    0;
   457,    204,    49,     8,    1,   0;
  4626,   2065,   496,    81,   10,   1,  0;
55969,  24984,  6001,   980,  121,  12,  1, 0;
788192, 351841, 84510, 13801, 1704, 169, 14, 1, 0;

Cf. A036242, A036243.

T := (n, k) -> BesselK(n,1)*BesselI(k,1) - (-1)^(n+k)*BesselI(n,1)* BesselK(k,1);
seq(lprint(seq(round(evalf(T(n, k), 99)), k=0..n)), n=0..8);

T = lambda n, k: bessel_K(n,1)*bessel_I(k,1) - (-1)^(n+k)*bessel_I(n,1)* bessel_K(k,1)
for n in range(9): [T(n,k).n().round() for k in (0..n)]

T(n, 0) = A036243(n-1) for n>=2.

T(n, 1) = A036242(n-1) for n>=2.

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

A036243 Denominator of fraction equal to the continued fraction [ 0, 2, 4, ...2n ].

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A246658 Triangle read by rows: T(n,k) = K(n,1)I(k,1) - (-1)^(n+k)I(n,1)* K(k,1), where I(n,x) and K(n,x) are Bessel functions; 0<=k<=n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Sage

Formula

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

A036243 Denominator of fraction equal to the continued fraction [ 0, 2, 4, ...2n ].

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A246658 Triangle read by rows: T(n,k) = K(n,1)*I(k,1) - (-1)^(n+k)*I(n,1)* K(k,1), where I(n,x) and K(n,x) are Bessel functions; 0<=k<=n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Sage

Formula

A246658 Triangle read by rows: T(n,k) = K(n,1)I(k,1) - (-1)^(n+k)I(n,1)* K(k,1), where I(n,x) and K(n,x) are Bessel functions; 0<=k<=n.