Geodesic
The Pfaffian transform
Journal of integer sequences, Tome 12 (2009) no. 1
We introduce a function on sequences, which we call the Pfaffian transform, using the Pfaffian of a skew-symmetric matrix. We establish several basic properties of the Pfaffian transform, and we use the transfer matrix method to show that the set of sequences with rational generating functions is closed under the Pfaffian transform. We conclude by computing the Pfaffian transform of a variety of sequences, including geometric sequences, the sequence of Fibonacci numbers, the sequence of Pell numbers, the sequence of Jacobsthal numbers, and the sequence of Tribonacci numbers. Throughout we describe a generalization of our results to Pfaffians of skew-symmetric matrices whose entries satisfy a Pascal-like relation.
Classification : 15A15, 05C50, 05C70
Keywords: Pfaffian, skew-symmetric matrix, Toeplitz matrix, transfer matrix method, perfect matching, Hankel transform 
@article{JIS_2009__12_1_a1,
     author = {Austin,  Tracale and Bantilan,  Hans and Egge,  Eric S. and Jonas,  Isao and Kory,  Paul},
     title = {The {Pfaffian} transform},
     journal = {Journal of integer sequences},
     year = {2009},
     volume = {12},
     number = {1},
     zbl = {1228.44004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2009__12_1_a1/}
}
TY  - JOUR
AU  - Austin,  Tracale
AU  - Bantilan,  Hans
AU  - Egge,  Eric S.
AU  - Jonas,  Isao
AU  - Kory,  Paul
TI  - The Pfaffian transform
JO  - Journal of integer sequences
PY  - 2009
VL  - 12
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/JIS_2009__12_1_a1/
LA  - en
ID  - JIS_2009__12_1_a1
ER  - 
%0 Journal Article
%A Austin,  Tracale
%A Bantilan,  Hans
%A Egge,  Eric S.
%A Jonas,  Isao
%A Kory,  Paul
%T The Pfaffian transform
%J Journal of integer sequences
%D 2009
%V 12
%N 1
%U http://geodesic.mathdoc.fr/item/JIS_2009__12_1_a1/
%G en
%F JIS_2009__12_1_a1
Austin,  Tracale; Bantilan,  Hans; Egge,  Eric S.; Jonas,  Isao; Kory,  Paul. The Pfaffian transform. Journal of integer sequences, Tome 12 (2009) no. 1. http://geodesic.mathdoc.fr/item/JIS_2009__12_1_a1/