A066224 -id:A066224 - cp's OEIS frontend

A130757 Triangular table of coefficients of Laguerre-Sonin polynomials n!2^nLag(n,x/2,1/2) of order 1/2.

1, 3, -1, 15, -10, 1, 105, -105, 21, -1, 945, -1260, 378, -36, 1, 10395, -17325, 6930, -990, 55, -1, 135135, -270270, 135135, -25740, 2145, -78, 1, 2027025, -4729725, 2837835, -675675, 75075, -4095, 105, -1, 34459425, -91891800, 64324260, -18378360, 2552550, -185640, 7140, -136

Offset: 0

Wolfdieter Lang, Jul 13 2007

These polynomials appear in the radial l=0 (s) wave functions of the isotropic three-dimensional harmonic quantum oscillator with the dimensionless variable x=(r/L)^2 with r>=0 and L^2=h/(m*f0). h is Planck's constant and m and f0 are the mass and the frequency of the oscillator.
From Tom Copeland, Dec 13 2015: (Start)
See A099174 for relations to the Hermite polynomials and the link in A176230 for operator relations. The infinitesimal generator for this matrix contains A014105.
The row polynomials are P(n,x) = 2^n n! Lag(n,x/2,1/2), where Lag(n,x,q) is the associated Laguerre polynomial of order q, with raising operator R = -x^(-2) [x^(3/2) (1 - 2D)]^2 = 3 - x + (4x - 6) D - 4x D^2 with D = d/dx, i.e., R P(n,x) - P(n+1,x). A matrix reresentation of R acting on an o.g.f. (formal power series) is given by the transpose of the production matrix below. The diagonal corresponds to (3 + 4 xD) x^n = (3 + 4n) x^n; the upper diagonal, to -x x^n = -x^(n+1); and the lower diagonal, to (-6 - 4 xD) D x^n = -n (6 + 4(n-1)) x^(n-1), the sequence A002943. See A176230 for a similar relation.
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.
(End)
Exponential Riordan array [1/(1-2x)^(3/2), -x/(1-2x)]. - Paul Barry, Mar 07 2017

			[1]; [3,-1]; [15,-10,1]; [105,-105,21,-1]; [945,-1260,378,-36,1]; ...

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 775, 22.3.9.
T. Copeland, Juggling Zeros in the Matrix: Example II, 2020.
Wolfdieter Lang, First ten rows and more.

Cf. A021009 (Coefficient table of n!*L(n, 0, x)).
Row sums (signed) give A131441. Row sums (unsigned) give A066224.
Cf. A001147, A002943, A014105, A049403, A066325, A096713, A099174, A100861, A104556, A111924, A122848, A144299, A176230.

seq(seq(n!*2^(n-m)*(-1)^m*binomial(n+1/2,n-m)/m!,m=0..n),n=0..10); # Robert Israel, Dec 25 2015

Table[n! (2^(n - m)) ((-1)^m) Binomial[n + 1/2, n - m]/m!, {n, 0, 8}, {m, 0, n}] // Flatten (* Michael De Vlieger, Dec 24 2015 *)

a(n,m) = n!*(2^(n-m))*L(1/2,n,m) with L(1/2,n,m) = ((-1)^m)*binomial(n+1/2,n-m)/m!, n >= m >= 0, otherwise 0.
Let IP be the lower triangular matrix with its first subdiagonal equal to the first subdiagonal (cf. A014105) of this entry's unsigned matrix M and with all other elements equal to zero. Then IP is the infinitesimal generator of M, i.e., M = exp(IP). - Tom Copeland, Dec 12 2015
From Tom Copeland, Dec 14 2015: (Start)
Production matrix is
   3,   -1;
  -6,    7,   -1;
   0,  -20,   11,   -1;
   0,    0,  -42,   15,   -1;
   0,    0,    0,  -72,   19,   -1;
   0,    0,    0,    0, -110,   23,   -1;
   0,    0,    0,    0,    0, -156,   27,   -1;
   0,    0,    0,    0,    0,    0, -210,   31,   -1;
   0,    0,    0,    0,    0,    0,    0, -272,   35,   -1;
  ... (End)

Title formula corrected by Tom Copeland, Dec 12 2015

A066223 Bisection of A000085.

Original entry on oeis.org

1, 2, 10, 76, 764, 9496, 140152, 2390480, 46206736, 997313824, 23758664096, 618884638912, 17492190577600, 532985208200576, 17411277367391104, 606917269909048576, 22481059424730751232, 881687990282453393920, 36494410645223834692096, 1589659519990672490875904

Offset: 0

N. J. A. Sloane, Dec 19 2001

Number of tableaux on 2n elements. - Roberto E. Martinez II, Jan 09 2002
a(n) = number of ways to connect 2n points labeled 1,2,...,2n in a line with 0 or more arcs such that at most one arc leaves each point. For example, with arcs separated by dashes, a(2)=10 counts {} (no arcs), 12, 13, 14, 23, 24, 34, 12-34, 13-24, 14-23. - David Callan, Sep 18 2007
a(n) = A229223(2n,2) = A229243(2,n). - Alois P. Heinz, Sep 17 2013

S. Chowla, The asymptotic behavior of solutions of difference equations, in Proceedings of the International Congress of Mathematicians (Cambridge, MA, 1950), Vol. I, 377, Amer. Math. Soc., Providence, RI, 1952.

Alois P. Heinz, Table of n, a(n) for n = 0..200
T. Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
I. Dolinka, J. East and R. D. Gray, Motzkin monoids and partial Brauer monoids, arXiv preprint arXiv:1512.02279 [math.GR], 2015.
Michael Torpey, Semigroup congruences: computational techniques and theoretical applications, Ph.D. Thesis, University of St. Andrews (Scotland, 2019).

Cf. A066224.

a:= proc(n) option remember; `if`(n<2, n+1,
      (4*n-2)*a(n-1)-2*(n-1)*(2*n-3)*a(n-2))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 17 2013

NumberOfTableaux[2n]
a[n_] := a[n] = If[n<2, n+1, (4*n-2)*a[n-1] - 2*(n-1)*(2*n-3)*a[n-2]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 13 2014, after Alois P. Heinz *)
Table[(-2)^n HypergeometricU[-n, 1/2, -(1/2)], {n, 0, 90}] (* Emanuele Munarini, Aug 31 2017 *)

a(n)=sum(k=0,n,binomial(2*n,2*k)*prod(i=1,k,2*i-1))

a(n)=if(n<0, 0, n*=2; n!*polcoeff(exp(x+x^2/2+x*O(x^n)),n))

a(n) = sum(k=0, n, C(2n, 2*k)*(2k-1)!!). - Benoit Cloitre, May 01 2003
a(n) = n!*2^n*LaguerreL(n, -1/2, -1/2). - Vladeta Jovovic, May 10 2003
E.g.f.: cosh(x)*exp(x^2/2) (with interpolated zeros) - Paul Barry, May 26 2003
E.g.f.: exp(x/(1-2*x))/sqrt(1-2*x). - Paul Barry, Apr 12 2010
a(n) = (1/sqrt(2*pi))*Int((1+x)^(2*n)*exp(-x^2/2),x,-infinity,infinity). - Paul Barry, Apr 21 2010
Conjecture: a(n) +2*(-2*n+1)*a(n-1) +2*(n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 24 2012
Remark: the above conjectured recurrence is true and can be obtained by the e.g.f. - Emanuele Munarini, Aug 31 2017
a(n) ~ n^n*2^(n-1/2)*exp(-n+sqrt(2*n)-1/4) * (1 + 7/(24*sqrt(2*n))). - Vaclav Kotesovec, Jun 22 2013

More terms from Roberto E. Martinez II, Jan 09 2002

cp's OEIS Frontend

A130757 Triangular table of coefficients of Laguerre-Sonin polynomials n!2^nLag(n,x/2,1/2) of order 1/2.

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cp's OEIS Frontend

A130757 Triangular table of coefficients of Laguerre-Sonin polynomials n!*2^n*Lag(n,x/2,1/2) of order 1/2.

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A066223 Bisection of A000085.

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A130757 Triangular table of coefficients of Laguerre-Sonin polynomials n!2^nLag(n,x/2,1/2) of order 1/2.