Clear["Global`*"];

p=4;k=4;  (*  set these both to the order of  the formula desired  *)

(* Adams-Bashforth Predictor Coefficients *)

eqns=Table[Sum[a[k-i]If[i==0&&j==0,1,(i^j)]/j!+b[k-i]If[i==0&&j==1,1,(i^((j-1)))]/(j-1)!,{i,0,k}]==0,{j,1,p}]/.Join[{b[0]->0,a[0]->-1,a[1]->1},Table[a[j]->0,{j,2,p}]]

Solve[eqns]

(* Adams-Moulton Corrector Coefficients *)

eqns=Table[Sum[a[k-i]If[i==0&&j==0,1,(i^j)]/j!+c[k-i]If[i==0&&j==1,1,(i^((j-1)))]/(j-1)!,{i,0,k}]==0,{j,1,p}]/.Join[{c[p]->0,a[0]->-1,a[1]->1},Table[a[j]->0,{j,2,p}]]

Solve[eqns]

(*  the following is another way that may be easier to understand  *)

Clear["Global`*"]

(*  first, we get the formula for the Adams-Bashforth predictor  *)

n=4;   (*  the order of the formula we want  *)

values=Table[{t-k h,f[t-k h]},{k,0,n-1}]

(*  now, we get the polynomial that interpolates the values of f  *)

poly[s_]:=InterpolatingPolynomial[values,s]

(*  now, we integrate the polynomial from t to t+h  *)

intpoly=(∫_t^(t+h))poly[s]\[DifferentialD]s

(*  and we can collect the terms that multiply h  *)

intpolyc=Collect[intpoly,h]

(*  now just get a common denominator (like the textbook does)  *)

Together[intpolyc]

(*  now, we try to make it look like the formula in (2) p. 347  *)

predictorFormula=x[t+h]==x[t]+ Together[intpolyc]

(*  in other words, this is what the formula in the text says  *)

"  x[t+h]==x[t]+(1/24) h (55 f[x(t)]-59 f[x(t-h)]+37 f[x(t-2 h)]-9 f[x(t-3 h)])  "

Clear["Global`*"]

(*  first, we get the formula for the Adams-Moulton correcter  *)

n=4;   (*  the order of the formula we want  *)

values=Table[{t-k h,f[t-k h]},{k,-1,n-2}]

(*  now, we get the polynomial that interpolates the values of f  *)

poly[s_]:=InterpolatingPolynomial[values,s]

(*  now, we integrate the polynomial from t to t+h  *)

intpoly=(∫_t^(t+h))poly[s]\[DifferentialD]s

(*  and we can collect the terms that multiply h  *)

intpolyc=Collect[intpoly,h]

(*  now just get a common denominator (like the textbook does)  *)

Together[intpolyc]

(*  now, we try to make it look like the formula in (3) on p. 348  *)

correctorFormula=x[t+h]==x[t]+ Together[intpolyc]

(*  in other words, this is what the formula in the text says  *)

"  x[t+h]==x[t]+(1/24) h (9 f[x(t+h)]+19 f[x(t)]-5 f[x(t-h)]+f[x(t-2 h)])  "