%I #17 Jul 11 2018 06:39:14

%S 1,3,5,105,189,3465,19305,2027025,3828825,130945815,1249937325,

%T 105411381075,608142583125,30494006668125,412685556908625,

%U 191898783962510625,372509404162520625,24627010608522196875

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

%C The denominators are A130188.

%C 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).

%C 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.

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

%H G. C. Greubel, <a href="/A130187/b130187.txt">Table of n, a(n) for n = 0..403</a>

%H Wolfdieter Lang, <a href="/A130187/a130187.pdf">Rationals and z-sequence.</a>

%F 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.

%F 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

%e r(1)=3/4 leads to z(3)=-3/8.

%e Rationals r(n):

%e 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 - ...

%e z-sequence: [0, -1/2, 0, -3/8, 0, -5/8, 0, -105/64, 0, -189/32, 0, ...]

%e Recurrence, n=4: H(4,0) = 4*(z(1)*(-12) + z(3)*8) = 4*((-1/2)*(-12) + (-3/8)*8) = 4*3 = 12.

%e 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

%t 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 *)

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_ Jun 01 2007