A025164 -id:A025164 - 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

A176230 Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].

Original entry on oeis.org

1, 1, 1, 3, 6, 1, 15, 45, 15, 1, 105, 420, 210, 28, 1, 945, 4725, 3150, 630, 45, 1, 10395, 62370, 51975, 13860, 1485, 66, 1, 135135, 945945, 945945, 315315, 45045, 3003, 91, 1, 2027025, 16216200, 18918900, 7567560, 1351350, 120120, 5460, 120, 1, 34459425

Offset: 0

Paul Barry, Apr 12 2010

Row sums are A066223. Reverse of A119743. Inverse is alternating sign version.
Diagonal sums are essentially A025164.
From Tom Copeland, Dec 13 2015: (Start)
See A099174 for relations to the Hermite polynomials and the link for operator relations, including the infinitesimal generator containing A000384.
Row polynomials are 2^n n! Lag(n,-x/2,-1/2), where Lag(n,x,q) is the associated Laguerre polynomial of order q.
The triangles of Bessel numbers entries A122848, A049403, A096713, A104556 contain these polynomials as even or odd rows. Also the aerated version A099174 and A066325. Reversed, these entries are A100861, A144299, A111924.
Divided along the diagonals by the initial element (A001147) of the diagonal, this matrix becomes the even rows of A034839.
(End)
The first few rows appear in expansions related to the Dedekind eta function on pp. 537-538 of the Chan et al. link. - Tom Copeland, Dec 14 2016

			Triangle begins
        1,
        1,        1,
        3,        6,        1,
       15,       45,       15,       1,
      105,      420,      210,      28,       1,
      945,     4725,     3150,     630,      45,      1,
    10395,    62370,    51975,   13860,    1485,     66,    1,
   135135,   945945,   945945,  315315,   45045,   3003,   91,   1,
  2027025, 16216200, 18918900, 7567560, 1351350, 120120, 5460, 120, 1
Production matrix is
  1,  1,
  2,  5,  1,
  0, 12,  9,  1,
  0,  0, 30, 13,  1,
  0,  0,  0, 56, 17,   1,
  0,  0,  0,  0, 90,  21,   1,
  0,  0,  0,  0,  0, 132,  25,   1,
  0,  0,  0,  0,  0,   0, 182,  29,  1,
  0,  0,  0,  0,  0,   0,   0, 240, 33, 1.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Bala, Generalized Dobinski formulas
H. Chan, S. Cooper, and P. Toh, The 26th power of Dedekind's eta function Advances in Mathematics, 207 (2006) 532-543.
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras, 2012.
Tom Copeland, Juggling Zeros in the Matrix (Example II), 2020.

Cf. A113278.

Cf. A000384, A001147, A034839, A049403, A066325, A096713, A099174, A100861, A104556, A111924, A122848, A144299.

ser := n -> series(KummerU(-n, 1/2, x), x, n+1):
seq(seq((-2)^(n-k)*coeff(ser(n), x, k), k=0..n), n=0..8); # Peter Luschny, Jan 18 2020

t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); u[n_, k_] := t[2 n, k + n]; Table[ u[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jan 14 2011 *)

Number triangle T(n,k) = (2n)!/((2k)!(n-k)!2^(n-k)).
T(n,k) = A122848(2n,k+n). - R. J. Mathar, Jan 14 2011
[x^(1/2)(1+2D)]^2 p(n,x)= p(n+1,x) and [D/(1+2D)]p(n,x)= n p(n-1,x) for the row polynomials of T, with D=d/dx. - Tom Copeland, Dec 26 2012
E.g.f.: exp[t*x/(1-2x)]/(1-2x)^(1/2). - Tom Copeland , Dec 10 2013
The n-th row polynomial R(n,x) is given by the type B Dobinski formula R(n,x) = exp(-x/2)*Sum_{k>=0} (2*k+1)*(2*k+3)*...*(2*k+1+2*(n-1))*(x/2)^k/k!. Cf. A113278. - Peter Bala, Jun 23 2014
The raising operator in my 2012 formula expanded is R = [x^(1/2)(1+2D)]^2 = 1 + x + (2 + 4x) D + 4x D^2, which in matrix form acting on an o.g.f. (formal power series) is the transpose of the production array below. The linear term x is the diagonal of ones after transposition. The main diagonal comes from (1 + 4xD) x^n = (1 + 4n) x^n. The last diagonal comes from (2 D + 4 x D^2) x^n = (2 + 4 xD) D x^n = n * (2 + 4(n-1)) x^(n-1). - Tom Copeland, Dec 13 2015
T(n, k) = (-2)^(n-k)*[x^k] KummerU(-n, 1/2, x). - Peter Luschny, Jan 18 2020

A036244 Denominator of continued fraction given by C(n) = [ 1; 3, 5, 7, ...(2n-1)].

Original entry on oeis.org

1, 3, 16, 115, 1051, 11676, 152839, 2304261, 39325276, 749484505, 15778499881, 363654981768, 9107153044081, 246256787171955, 7150553981030776, 221913430199126011, 7330293750552189139, 256782194699525745876, 9508271497633004786551, 371079370602386712421365

Offset: 1

Jeff Burch

Denominators of convergents to coth(1) = 1.313035... = A073747.

Convergents: 1/1, 4/3, 21/16, 151/115, ... - Michael Somos, Sep 27 2017

			G.f. = x + 3*x^2 + 16*x^3 + 115*x^4 + 1051*x^5 + 11676*x^6 + 152839*x^7 + ...

Robert Israel, Table of n, a(n) for n = 1..369

Cf. A001040, A001053, A073747.

Numerators are sequence A025164. A058798.

I:=[1,3]; [n le 2 select I[n] else (2*n-1)*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Apr 19 2015

seq(denom(numtheory:-cfrac([seq(2*i-1,i=1..n)])),n=1..50); # Robert Israel, Apr 19 2015

Rest[CoefficientList[Series[(E^(1-(1-2*x)^(1/2))/2 - E^(-1+(1-2*x)^(1/2))/2) / (1-2*x)^(1/2), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Oct 05 2013 *)
a[ n_ ] := a[n] =a[n-2]+(2 n-1) a[n-1]; a[0] := 0; a[1] := 1.  RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-2]+(2n-1)a[n-1]}, a, {n, 20}] (* G. C. Greubel,Apr 23 2015 *)
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, Sep 27 2017 *)
Table[FromContinuedFraction[Range[1,2n+1,2]],{n,0,20}]//Denominator (* Harvey P. Dale, May 06 2018 *)
Convergents[Coth[1], 20] // Denominator (* Jean-François Alcover, Jun 15 2019 *)

def A036244(n):
    if n == 1: return 1
    return 2^n*gamma(n+1/2)*hypergeometric([1/2-n/2, 1-n/2], [3/2, 1/2-n, 1-n], 1)/sqrt(pi)
[round(A036244(n).n(100)) for n in (1..20)] # Peter Luschny, Sep 11 2014

a(n) = a(n-1)*(2*n-1) + a(n-2); a(0) = 0, a(1) = 1.
E.g.f.: sinh(1-(1-2*x)^(1/2))/(1-2*x)^(1/2). - Vladeta Jovovic, Jan 30 2004
E.g.f.: cosh(1-(1-2*x)^(1/2))/(1-2*x) + sinh(1-(1-2*x)^(1/2))/((1-2*x)^(3/2)).
E.g.f. G(0)/(1-2*x) where G(k)= 1 + 2*x/((2*k+1)*(1-2*x+sqrt(1-2*x))+(2*k+1)*(4*x^2-2*x)/(-1+2*x+sqrt(1-2*x) + (2*k+2)/G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Jul 01 2012
a(n) = Sum_{k=0..floor((n-1)/2)} 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) ~ (exp(2)-1)*2^(n-1/2)*n^n/exp(n+1). - Vaclav Kotesovec, Oct 05 2013
a(n) = A001147(n)*hypergeometric([1/2-n/2, 1-n/2], [3/2, 1/2-n, 1-n], 1) for n >= 2. - Peter Luschny, 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 (where a(0) = 0). - G. C. Greubel, Apr 18 2015
a(n) = A025164(-1-n) for all n in Z. - Michael Somos, Sep 27 2017

More terms from Larry Reeves (larryr(AT)acm.org), May 15 2001
More terms from Benoit Cloitre, Dec 20 2002
More terms from Vladeta Jovovic, Jan 30 2004

A176231 Coefficient array of orthogonal polynomials whose moment sequence is the double factorial numbers A001147.

Original entry on oeis.org

1, -1, 1, 3, -6, 1, -15, 45, -15, 1, 105, -420, 210, -28, 1, -945, 4725, -3150, 630, -45, 1, 10395, -62370, 51975, -13860, 1485, -66, 1, -135135, 945945, -945945, 315315, -45045, 3003, -91, 1, 2027025, -16216200, 18918900, -7567560, 1351350, -120120, 5460, -120, 1

Offset: 0

Paul Barry, Apr 12 2010

Exponential Riordan array [1/sqrt(1+2x),x/(1+2x)]. Inverse of A176230.

Diagonal sums are an alternating sign version of A025164.

			Triangle begins
  1,
  -1, 1,
  3, -6, 1,
  -15, 45, -15, 1,
  105, -420, 210, -28, 1,
  -945, 4725, -3150, 630, -45, 1,
  10395, -62370, 51975, -13860, 1485, -66, 1,
  -135135, 945945, -945945, 315315, -45045, 3003, -91, 1,
  2027025, -16216200, 18918900, -7567560, 1351350, -120120, 5460, -120, 1
Production matrix is
  -1, 1,
  2, -5, 1,
  0, 12, -9, 1,
  0, 0, 30, -13, 1,
  0, 0, 0, 56, -17, 1,
  0, 0, 0, 0, 90, -21, 1,
  0, 0, 0, 0, 0, 132, -25, 1,
  0, 0, 0, 0, 0, 0, 182, -29, 1,
  0, 0, 0, 0, 0, 0, 0, 240, -33, 1

Cf. A001147, A025164, A176230.

T := (n,k) -> (2*n)!*(-1/2)^(n-k)/(2*k)!*(n-k)!:
seq(seq(T(n,k), k=0..n), n=0..8); # Peter Luschny, Jul 20 2019

(* The function RiordanArray is defined in A256893. *)
rows = 9;
R = RiordanArray[1/Sqrt[1 + 2 #]&, #/(1 + 2 #)&, rows, True];
R // Flatten (* Jean-François Alcover, Jul 20 2019 *)
T[ n_, k_] := Coefficient[ HermiteH[2 n, x/Sqrt[2]], x, 2 k]/2^n; (* Michael Somos, Jan 15 2020 *)
T[ n_, k_] := Coefficient[ Nest[# x - D[#, x]&, 1, 2 n], x, 2 k]; (* Michael Somos, Jan 15 2020 *)

{T(n, k) = my(t=1); for(i=1, 2*n, t = x*t - t'); polcoeff(t, 2*k)}; /* Michael Somos, Jan 15 2020 */

Number triangle T(n,k) = (-1)^(n-k)*(2n)!/((2k)!(n-k)!2^(n-k)).

He_(2*n)(x) = Sum_{k=0..n} T(n, k)*x^(2*k) where He is Hermite's polynomial. - Michael Somos, Jan 15 2020

cp's OEIS Frontend

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

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A176230 Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].

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A036244 Denominator of continued fraction given by C(n) = [ 1; 3, 5, 7, ...(2n-1)].

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Magma

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A176231 Coefficient array of orthogonal polynomials whose moment sequence is the double factorial numbers A001147.

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Author

Keywords

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Examples

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Programs

Maple

Mathematica

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Formula