|
|
A292220
|
|
Expansion of the exponential generating function (1/2)*(1 + 4*x)*(1 - (1 + 4*x)^(-1/2))/x.
|
|
3
|
|
|
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
|
|
|
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
|
|
EXAMPLE
|
The sequence z(4,1;n) = 2*a(n) begins: {2,2,-8,60,-672,10080,-190080,4324320,-115315200,3528645120,-121898649600,...}.
|
|
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):
|
|
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 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|