A292220 Expansion of the exponential generating function (1/2)*(1 + 4*x)*(1 - (1 + 4*x)^(-1/2))/x.

1, 1, -4, 30, -336, 5040, -95040, 2162160, -57657600, 1764322560, -60949324800, 2346549004800, -99638080819200, 4626053752320000, -233153109116928000, 12677700308232960000, -739781100339240960000, 46113021921146019840000, -3058021453718104473600000

Offset: 0

Author

Wolfdieter Lang, Sep 13 2017

Keywords

• sign
• easy

Comments

This gives one half of the z-sequence entries for the generalized unsigned Lah number Sheffer matrix Lah[4,1] = A048854.

For Sheffer a- and z-sequences see a W. Lang link under A006232 with the references for the Riordan case, and also the present link for a proof.

Examples

			The sequence z(4,1;n) = 2*a(n) begins: {2,2,-8,60,-672,10080,-190080,4324320,-115315200,3528645120,-121898649600,...}.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..366
• Wolfdieter Lang, Note on a- and z-sequences of Sheffer number triangles for certain generalized Lah numbers.

Crossrefs

Cf. A001147, A006232 (link), A048854, A292221 (z[4,3]/2).

Programs

Maple

f:= gfun:-rectoproc({a(n+1) = -2*(1 + 2*n)*(1 + n)*a(n)/(2 + n),a(0)=1,a(1)=1},a(n),remember):
map(f, [$0..30]); # Robert Israel, May 10 2020

Mathematica

With[{nn=20},CoefficientList[Series[1/2 (1+4x) (1-(1+4x)^(-1/2))/x,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Aug 01 2021 *)

Formula

a(n) = [x^n/n!] (1/2)*(1 + 4*x)*(1 - (1 + 4*x)^(-1/2))/x.

a(0) = 1, a(n) = -(-2)^n*Product_{j=1..n} (2*j - 1)/(n+1) = -((-2)^n/(n+1))*A001147(n), n >= 1.

a(n) ~ -(-1)^n * n^(n-1) * 2^(2*n + 1/2) / exp(n). - Vaclav Kotesovec, Sep 18 2017

a(n+1) = -2*(1 + 2*n)*(1 + n)*a(n)/(2 + n) for n >= 1. - Robert Israel, May 10 2020