[seqfan] A101428 compared to A101409, polyminoes and trees

[Richard Mathar]

Just by comparing numbers it seems that A101428 and A101409 are the same.
Also by evaluation of the two different generating functions (see the Maple
code below), individually along the rows and columns, we get the same arrays.
  However, I do not manage to demonstrate equality by converting the two
generating functions in analytical terms (?) There ought be somewhere a trick
(math is a science, let's call it a methodology then) to convert some property
of the g-term (inverse trigonometric sine essentially) to establish this with
rigor...

http://research.att.com/~njas/sequences/?q=id:A101428|id:A101409

A101428 :=proc(n,k)
        g:=2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z):
        G28:= (1-t*z*g^2)/(1-t*z*g-t*z*g^2):
        coeftayl(G28,z=0,n) ;
        coeftayl(%,t=0,k) ;
end:

A101409 :=proc(n,k)
        g:=2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z):
        G09:= t*z*g/(1-t*z*g-t*z*g^2):
        coeftayl(G09,z=0,n) ;
        coeftayl(%,t=0,k) ;
end:

for n from 1 to 10 do
        for k from 1 to n do
                print(n,k,A101428(n,k),A101409(n,k)) ;
        od:
od:

Richard J. Mathar