Gottfried,
       As usual, you have some very nice insights. 
Glad to supply that table (to be submitted as A120013) below (15x15).
Notice that the first negative term in each row n occurs at about k=n+3
Thanks for the interest! 
     Paul 
 

{[1,1,1,0,-6,-33,-143,-572,-2210,-8398,-31654,-118864,-445740,-1671525,-6273135],
[1,2,4,6,-4,-100,-664,-3514,-16916,-77388,-343144,-1490148,-6376616,-26992264,-113317936],
[1,3,9,24,42,-87,-1575,-12240,-77730,-449994,-2470278,-13101228,-67823484,-344888619,-1729791975],
[1,4,16,60,192,360,-1568,-26804,-240800,-1804456,-12392608,-80842408,-509753856,-3137632560,-18967622976],
[1,5,25,120,530,1955,3785,-28900,-508610,-5227110,-45465910,-364121520,-2775443300,-20469321505,-147449975575],
[1,6,36,210,1164,5892,24552,48258,-577380,-10814388,-125790504,-1250747244,-11489717640,-100612975800,-853199104560],
[1,7,49,336,2226,13965,79681,370216,733054,-12716578,-256589494,-3342818668,-37486215396,-389327197959,-3858887809791],
[1,8,64,504,3872,28688,200960,1276760,6548320,13015536,-309382528,-6745255248,-97479894464,-1218641005024,-14135197035520],
[1,9,81,720,6282,53415,437697,3387636,23729310,133234434,265377402,-8284701168,-194970467196,-3098840425245,-42763998536991],
[1,10,100,990,9660,92460,862120,7743550,65644780,502780580,3068613080,6119742860,-242846773800,-6152118566760,-106732661972400],
[1,11,121,1320,14234,151217,1575497,15987488,156229150,1440018822,11974950746,78960669876,157595733460,-7746814711075,-210546506575735],
[1,12,144,1716,20256,236280,2714976,30591396,335668224,3545900232,35314018656,316917570312,2245647762624,4484375206896,-267431828575296],
[1,13,169,2184,28002,355563,4461145,55133260,668095870,7882889066,89524393322,958109355632,9230453871756,69959441864871,139753495019385],
[1,14,196,2730,37772,518420,7046312,94629626,1251725020,16233723324,204975068312,2490640448772,28508766348664,293492484431720,2369496643448144],
[1,15,225,3360,49890,735765,10763505,155928600,2231876670,31463969070,434868620970,5851065721140,75747398563500,923461269689985,10117826259791025]}


-- Gottfried Helms <Annette.Warlich@t-online.de> wrote:
Paul -
 just for couriosity. I did a gauss-elimination
 at the matrix of your values at the end of your
 posting (where a(n) is then the principal diagonal only).
[...]
To have a deeper look at this I would like to
 have more coefficients for A. Would you mind
 to provide more of these coefficients, day 8 x 8
 or 12 by 12?

Regards -
Gottfried Helms

Am 09.06.2006 07:27 schrieb Paul D. Hanna:
> Max, and Seqfans, 
>     Using the lovely formula for the n-th term of the m-th power of
> Catalan function: 
> [x^n] Catalan(x)^m = m*C(2*n+m-1,n)/(n+m), 
> I have found the simplest form yet for A120014: 
>  

