A130187 Numerators of rationals r(n) related to the z-sequence of the Sheffer matrix A060821 for Hermite polynomials.

1, 3, 5, 105, 189, 3465, 19305, 2027025, 3828825, 130945815, 1249937325, 105411381075, 608142583125, 30494006668125, 412685556908625, 191898783962510625, 372509404162520625, 24627010608522196875

Offset

0,2

Comments

The denominators are A130188.

The z-sequence for the Sheffer matrix (see the W. Lang link under A006233) A060821(n,m) (coefficients of Hermite polynomials) is z(2*k)=0 and z(2*k+1) = -r(k)/2, k >= 0, with r(n) := a(n)/A130188(n).

The recurrence for the entries of the first (m=0) column of the Sheffer triangle A006233(n,m)=:H(n,m) is H(0,0):=1, H(n,0) = n*Sum_{m=0..n-1} z(m)*H(n-1, m), n >= 1.

The e.g.f. for the z-sequence is -2*(exp((x^2)/4)-1)/x.

Links

G. C. Greubel, Table of n, a(n) for n = 0..403

Wolfdieter Lang, Rationals and z-sequence.

Formula

a(n) = numerator(r(n)), n >= 0. r(n):=-2*z(2*n+1) (in lowest terms). The e.g.f. of z(n) is given above.

Conjecture: a(n) = A001147(n+1)/A000265(n+1), n >= 0. (Motivated to reconsider this sequence by an e-mail of Thomas Olson.) - Wolfdieter Lang, Jan 04 2013

Example

r(1)=3/4 leads to z(3)=-3/8.
Rationals r(n):
E.g.f. for z-sequence: -2*(exp((x^2)/4)-1)/x = -(1/2)*x - (1/16)*x^3 - (1/192)*x^5 - (1/3072)*x^7 - ...
z-sequence: [0, -1/2, 0, -3/8, 0, -5/8, 0, -105/64, 0, -189/32, 0, ...]
Recurrence, n=4: H(4,0) = 4*(z(1)*(-12) + z(3)*8) = 4*((-1/2)*(-12) + (-3/8)*8) = 4*3 = 12.
Conjecture checks: a(3) = A001147(4)/A000265(4) = 7!!/1 = 1*3*5*7 = 105. a(4) = A001147(5)/A000265(5) = 9!!/5 = 1*3*7*9 = 189. - Wolfdieter Lang, Jan 04 2013

Mathematica

F:= CoefficientList[Series[-2*(Exp[x^2/4] -1)/x, {x, 0, 75}], x]*Range[0, 75]!; Table[Numerator[-2*F[[2*n]]], {n, 1, 50}] (* G. C. Greubel, Jul 10 2018 *)

Crossrefs

Sequence in context: A346710 A234600 A003112 * A289488 A361092 A054266

Adjacent sequences: A130184 A130185 A130186 * A130188 A130189 A130190

Keyword

nonn,frac,easy

Author

Wolfdieter Lang Jun 01 2007

Status

approved