Generating Function of Riordan numbers I would like to find generating function of f(n), where f(n)

Joshua Foley 2022-07-08 Answered
Generating Function of Riordan numbers
I would like to find generating function of f(n), where f(n) is defined as following: f ( n ) = k n ( n k ) ( 1 ) n k C k .
With C k = 1 k + 1 ( 2 k k ) ( C k is the k t h Catalan's number).
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Answers (1)

SweallySnicles3
Answered 2022-07-09 Author has 21 answers
Step 1
We have for the sum
k = 0 n ( n k ) ( 1 ) n k C k = k = 0 n ( n k ) ( 1 ) k C n k = [ z n ] 1 1 4 z 2 z k = 0 n ( n k ) ( 1 ) k z k = [ z n ] 1 1 4 z 2 z ( 1 z ) n = r e s z 1 z n + 1 ( 1 z ) n 1 1 4 z 2 z .
Step 2
Now put z / ( 1 z ) = w so that z = w / ( 1 + w ) and d z = 1 / ( 1 + w ) 2 d w to find
r e s w 1 w n + 1 ( 1 + w ) 1 1 4 w / ( 1 + w ) 2 w / ( 1 + w ) 1 ( 1 + w ) 2 = r e s w 1 w n + 1 1 + w ( 1 + w ) 2 4 w ( 1 + w ) 2 w ( 1 + w ) = r e s w 1 w n + 1 1 + w 1 2 w 3 w 2 2 w ( 1 + w ) .
It follows that the desired OGF is 1 + w 1 2 w 3 w 2 2 w ( 1 + w ) ..

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