A093986 -id:A093986 - 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

A093985 a(1) = 1, a(2) = 2; a(n+1) = 2n*a(n) - a(n-1). Symmetrically, a(n) = (a(n-1) + a(n+1))/((n-1) + (n+1)).

Original entry on oeis.org

1, 2, 7, 40, 313, 3090, 36767, 511648, 8149601, 146181170, 2915473799, 63994242408, 1532946343993, 39792610701410, 1112660153295487, 33340011988163200, 1065767723467926913, 36202762585921351842, 1302233685369700739399, 49448677281462706745320

Offset: 1

Amarnath Murthy, May 22 2004

			a(3)=7 because 2*2*a(2) - a(1) = 7.

Seiichi Manyama, Table of n, a(n) for n = 1..404

Cf. A058798, A093986.

I:=[1,2]; [n le 2 select I[n] else 2*(n-1)*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 15 2017

a[1]:=1: a[2]:=2: for n from 2 to 21 do a[n+1]:=2*n*a[n]-a[n-1] od: seq(a[n],n=1..21); # Emeric Deutsch, Jul 31 2005

nxt[{n_,a_,b_}]:={n+1,b,2*n*b-a}; NestList[nxt,{2,1,2},20][[All,2]] (* Harvey P. Dale, Jan 09 2021 *)
a[n_] := (Pi/2)*(BesselY[0, 1]*BesselJ[n, 1.] - BesselJ[0, 1]*BesselY[n, 1.]);
Table[Round[a[n]], {n, 1, 20}] (* Hugo Pfoertner, Feb 12 2024 *)

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))^2. Cf. A058798. - Peter Bala, Aug 01 2013

a(n) = (Pi/2)*(Y[0, 1] * J[n, 1] - J[0, 1] * Y[n, 1]) where Y and J are Bessel functions. - Peter Luschny, Jan 30 2024

Corrected and extended by Emeric Deutsch, Jul 31 2005

A093987 a(1) = 1, a(2) = 2, a(n+1) = a(n)^(2n)/ a(n-1). Symmetrically a(n) = {a(n-1)*a(n+1)}^[1/{(n-1) +(n+1)}].

Original entry on oeis.org

1, 2, 16, 8388608, 1532495540865888858358347027150309183618739122183602176

Offset: 1

Amarnath Murthy, May 22 2004

nonn

The next term has 535 digits. - Harvey P. Dale, Jan 31 2020

			a(5) = 8388608^8/16 = 1.53249554e+54.
a(6) =~ 8.5173068734e+534.

Cf. A093985, A093986.

a[1] = 1; a[2] = 2; a[n_] := a[n - 1]^(2n - 2)/a[n - 2]; Table[ a[n], {n, 6}] (* Robert G. Wilson v, May 24 2004 *)
RecurrenceTable[{a[1]==1,a[2]==2,a[n+1]==a[n]^(2n)/a[n-1]},a,{n,5}] (* Harvey P. Dale, Jan 31 2020 *)

Edited and extended by Robert G. Wilson v, May 24 2004

A358735 Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (mx + 2)a(m+1) - a(m) for all m in Z.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 10, 16, 6, 1, 20, 70, 76, 24, 1, 35, 224, 496, 428, 120, 1, 56, 588, 2260, 3808, 2808, 720, 1, 84, 1344, 8140, 23008, 32152, 21096, 5040, 1, 120, 2772, 24772, 107328, 245560, 298688, 178848, 40320

Offset: 0

Michael Somos, Mar 15 2023

This sequence is essentially A204024 except for extra row, alternating signs and reversed rows.

The sequence of polynomials a(m) satisfies a(m)*a(m-2) = a(m-1) * (a(m-1) + x*a(m-2) + a(m-3)) - a(m-2)^2 for all m > 3.

			a(3) = 1, a(4) = 1 + x, a(5) = 1 + 4*x + 2*x^2.
Triangular array T(n, k) starts:
n\k | 0   1   2   3   4   5
--- + - --- --- --- --- ---
 0  | 1
 1  | 1   1
 2  | 1   4   2
 3  | 1  10  16   6
 4  | 1  20  70  76  24
 5  | 1  35 224 496 428 120

Cf. A000292, A040977, A058797, A093986, A204024.

T[ n_, k_] := If[ n<0, 0, Module[{a = Table[1, n+3], x}, Do[ a[[m]] = a[[m-1]] *(a[[m-1]] + x*a[[m-2]] + a[[m-3]])/a[[m-2]] - a[[m-2]] //Factor//Expand, {m, 4, n+3}]; Coefficient[ a[[n+3]], x, k]]];

{T(n, k) = if( n<0, 0, my(a = vector(n+3, i, 1)); for(m = 4, n+3, a[m] = a[m-1]*(a[m-1] + 'x*a[m-2] + a[m-3])/a[m-2] - a[m-2]); polcoeff( a[n+3], k))};

If x=1, then a(n) = A058797(n+2) = Sum_{k=0..n} T(n, k).
If x=2, then a(n) = A093986(n+2).
T(n, n) = n!, T(n, 0) = 1, T(n, 1) = A000292(n). T(n, 2) = 2*A040977(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

A093985 a(1) = 1, a(2) = 2; a(n+1) = 2n*a(n) - a(n-1). Symmetrically, a(n) = (a(n-1) + a(n+1))/((n-1) + (n+1)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A093987 a(1) = 1, a(2) = 2, a(n+1) = a(n)^(2n)/ a(n-1). Symmetrically a(n) = {a(n-1)*a(n+1)}^[1/{(n-1) +(n+1)}].

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A358735 Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (m*x + 2)*a(m+1) - a(m) for all m in Z.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A358735 Triangular array read by rows. T(n, k) is the coefficient of x^k in a(n+3) where a(1) = a(2) = a(3) = 1 and a(m+2) = (mx + 2)a(m+1) - a(m) for all m in Z.