[seqfan] Re: Definition of A002190

[Paul D Hanna]

Seqfans, 
      Setting the OFFSET=1 and including a(1)=1, then the sequence A002190 obeys the following 
FORMULA.
G.f.: 1 = Sum_{n>=0} a(n+1)*A000108(n)*x^n*Sum_{k>=0} C(2n+k,k)^2*(-x)^k.
Compare with the following g.f of the Catalan numbers (A000108):
  1 = Sum_{n>=0} A000108(n)*x^n*Sum_{k>=0} C(2n+k,k)*(-x)^k. 
 
EXAMPLE. 
The terms of this sequence can be generated by the g.f.:
1 = 1*1*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 +...)
+ 1*1*x*(1 - 3^2*x + 6^2*x^2 - 10^2*x^3 + 15^2*x^4 - 21^2*x^5 +...)
+ 4*2*x^2*(1 - 5^2*x + 15^2*x^2 - 35^2*x^3 + 70^2*x^4 - 126^2*x^5 +...)
+ 33*5*x^3*(1 - 7^2*x + 28^2*x^2 - 84^2*x^3 + 210^2*x^4 - 462^2*x^5+...)
+ 456*14*x^4*(1 - 9^2*x + 45^2*x^2 - 165^2*x^3 + 495^2*x^4 +...)
+ 9460*42*x^5*(1 - 11^2*x + 66^2*x^2 - 286^2*x^3 + 1001^2*x^4 +...)
+ 274800*132*x^6*(1 - 13^2*x + 91^2*x^2 - 455^2*x^3 + 1820^2*x^4 +...)
+ 10643745*429*x^7*(1 - 15^2*x + 120^2*x^2 - 680^2*x^3 + 3060^2*x^4 +...)
 +...
Compare to a g.f. of the Catalan numbers (A000108): 
1 = 1*(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 +...) 
+ 1*x*(1 - 3*x + 6*x^2 - 10*x^3 + 15*x^4 - 21*x^5 +...) 
+ 2*x^2*(1 - 5*x + 15*x^2 - 35*x^3 + 70*x^4 - 126*x^5 +...) 
+ 5*x^3*(1 - 7*x + 28*x^2 - 84*x^3 + 210*x^4 - 462*x^5 +...) 
+ 14*x^4*(1 - 9*x + 45*x^2 - 165*x^3 + 495*x^4 - 1287*x^5 +...) 
+ 42*x^5*(1 - 11*x + 66*x^2 - 286*x^3 + 1001*x^4 - 3003*x^5 +...) 
+ 132*x^6*(1 - 13*x + 91*x^2 - 455*x^3 + 1820*x^4 - 6188*x^5 +...) 
+... 
 
I have verified the above formula for the first 1000 terms ... 
can anyone prove it? 
     Paul