Please correct if there a mistake on my Explicit formula of 7th Steps Adam-Bashforth-Moulton

In the generic test case $y'=\lambda y$, you want in AB to approximate $\frac{e^{λh}-1}{λh}$ as a polynomial in $e^{-λh}$. Set $z=1-e^{-λh}$, then one has to solve $$ z=-\log(1-z)(1-z)q(z)+O(z^{p+1}) $$ CAS Magma (online calculator)

PS<z>:=PowerSeriesRing(Rationals());
for p in [2..9] do
    q:=-z/((1-z)*Log(1-z+O(z^(p+3))));
    q:=Truncate(q+O(z^p));
    c:=Coefficients(Evaluate(q, 1-z));
    den := LCM([Denominator(cc): cc in c]);
    Sprintf("%o: %o/%o", p, [ den*cc: cc in c],den);
end for;

giving the table of right sides

2: [ 3, -1 ]/2
3: [ 23, -16, 5 ]/12
4: [ 55, -59, 37, -9 ]/24
5: [ 1901, -2774, 2616, -1274, 251 ]/720
6: [ 4277, -7923, 9982, -7298, 2877, -475 ]/1440
7: [ 198721, -447288, 705549, -688256, 407139, -134472, 19087 ]/60480
8: [ 434241, -1152169, 2183877, -2664477, 2102243, -1041723, 295767, -36799 ]/120960
9: [ 14097247, -43125206, 95476786, -139855262, 137968480, -91172642, 38833486, -9664106, 1070017 ]/3628800

For Adams-Moulton you get that the highest indices on both sides are the same, so the task is now to approximate $\frac{1-e^{-λh}}{λh}$ with a polynomial in $e^{-λh}$ with remainder $O(h^p)$, that is, similarly to the above from

PS<z>:=PowerSeriesRing(Rationals());
for p in [2..9] do
    q:=-z/(Log(1-z+O(z^(p+3))));
    q:=Truncate(q+O(z^p));
    c:=Coefficients(Evaluate(q, 1-z));
    den := LCM([Denominator(cc): cc in c]);
    Sprintf("%o: %o/%o", p, [ den*cc: cc in c],den);
end for;

the table

2: [ 1, 1 ]/2
3: [ 5, 8, -1 ]/12
4: [ 9, 19, -5, 1 ]/24
5: [ 251, 646, -264, 106, -19 ]/720
6: [ 475, 1427, -798, 482, -173, 27 ]/1440
7: [ 19087, 65112, -46461, 37504, -20211, 6312, -863 ]/60480
8: [ 36799, 139849, -121797, 123133, -88547, 41499, -11351, 1375 ]/120960
9: [ 1070017, 4467094, -4604594, 5595358, -5033120, 3146338, -1291214, 312874, -33953 ]/3628800

Extracting specifically for order $p=7$, the coefficients are thus

AB: [ 198721, -447288, 705549, -688256, 407139, -134472, 19087 ]/60480
AM: [ 19087, 65112, -46461, 37504, -20211, 6312, -863 ]/60480

which show differences in several places to your result.