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A292220 Expansion of the exponential generating function (1/2)*(1 + 4*x)*(1 - (1 + 4*x)^(-1/2))/x. 1
1, 1, -4, 30, -336, 5040, -95040, 2162160, -57657600, 1764322560, -60949324800, 2346549004800, -99638080819200, 4626053752320000, -233153109116928000, 12677700308232960000, -739781100339240960000, 46113021921146019840000, -3058021453718104473600000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

LINKS

Table of n, a(n) for n=0..18.

Wolfdieter Lang, Note on a- and z-sequences of Sheffer number triangles for certain generalized Lah numbers.

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

EXAMPLE

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

CROSSREFS

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

Sequence in context: A207833 A121413 A001761 * A099712 A209440 A052316

Adjacent sequences:  A292217 A292218 A292219 * A292221 A292222 A292223

KEYWORD

sign,easy

AUTHOR

Wolfdieter Lang, Sep 13 2017

STATUS

approved

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Last modified December 14 17:37 EST 2018. Contains 318103 sequences. (Running on oeis4.)