The sequence OEIS A103904 is described as
Number of perfect matchings of an $n \times (n+1)$ Aztec rectangle with the third vertex in the topmost row removed.
Definition of $M \times N $ Aztec rectangle (one can see such a definition by C. Krattenthaler):
Consider a $(2M+1) \times (2N+1)$ rectangular chessboard and suppose that the corners are black. Then an $M \times N$ Aztec rectangle is the graph whose vertices are the white squares and whose edges connect precisely those pairs of white squares that are diagonally adjacent. So an $n \times (n+1)$ Aztec rectangle is when $M=n$ and $N=n+1$.
see the following figures for $n \times (n+1)$ Aztec rectangles as examples:
When $n=2$, I try to calculate the number of perfect matchings of $2 \times (2+1)$ Aztec rectangle with the third vertex in the topmost row removed:
The number of perfect matching is actually $8$, which is not in A103904. Is there any mistake in this sequence or the definition is wrong? Or maybe I misunderstand it. I am very happy to hear from anyone and thank you very much in advance!
@Marie, I agree with you that the sequence A103904 is mistaken.
The first term is plainly wrong (incidentally the formula section on the OEIS page has every entry marked with $n>1$, as none of them produces $1$ when $n=1$), but here are two ways to fix the rest:
(1) Keep the definition. The correct sequence is then given by $$ \frac{n(n-1)}{2}\times2^{n(n+1)/2}, $$ or $$ 0, 8, 192, 6144, 327680, \dots $$ i.e., the OEIS entry has the exponent wrong: it should be $n+1\choose2$ rather than $n\choose 2$, so the sequence is off by a factor of $2^n$. Note $n=1$ does not need to be singled out any more.
(2) Change the definition to the following:
Then the sequence can remain as it is, except the first term needs to be changed to $0$: $$ 0, 2, 24, 384, 10240, \dots $$
These results follow directly from Lemma 1 and 2 in Krattenthaler that you have quoted above. Let me know if you have any questions.