Mistake in OEIS A103904?

Solution 1:

@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:

Number of perfect matchings of an $n\times(n+1)$ Aztec rectangle, where all the vertices in the top-most row and the second top-most row have been removed, except the third and the last vertex of the second top-most row.

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.