s0=1;
s1=(1/2) (-2+α+β);
s2=(1/8) (4+3 (α^2)+2 α (-2+β)-4 β+3 (β^2));
s3=(1/16) (-8+5 (α^3)+3 (α^2) (-2+β)+4 β-6 (β^2)+5 (β^3)+α (4-4 β+3 (β^2)));
s4=(1/128) (48+35 (α^4)+20 (α^3) (-2+β)-32 β+24 (β^2)-40 (β^3)+35 (β^4)+6 (α^2) (4-4 β+3 (β^2))+4 α (-8+4 β-6 (β^2)+5 (β^3)));
s5=(1/256) (-96+63 (α^5)+35 (α^4) (-2+β)+48 β-48 (β^2)+40 (β^3)-70 (β^4)+63 (β^5)+10 (α^3) (4-4 β+3 (β^2))+6 (α^2) (-8+4 β-6 (β^2)+5 (β^3))+α (48-32 β+24 (β^2)-40 (β^3)+35 (β^4)));
s6=(1/1024)(320+231 (α^6)+126 (α^5) (-2+β)-192 β+144 (β^2)-160 (β^3)+140 (β^4)-252 (β^5)+231 (β^6)+35 (α^4) (4-4 β+3 (β^2))+20 (α^3) (-8+4 β-6 (β^2)+5 (β^3))+3 (α^2) (48-32 β+24 (β^2)-40 (β^3)+35 (β^4))+2 α (-96+48 β-48 (β^2)+40 (β^3)-70 (β^4)+63 (β^5)));
s7=(1/2048)(-640+429 (α^7)+231 (α^6) (-2+β)+320 β-288 (β^2)+240 (β^3)-280 (β^4)+252 (β^5)-462 (β^6)+429 (β^7)+63 (α^5) (4-4 β+3 (β^2))+35 (α^4) (-8+4 β-6 (β^2)+5 (β^3))+5 (α^3) (48-32 β+24 (β^2)-40 (β^3)+35 (β^4))+3 (α^2) (-96+48 β-48 (β^2)+40 (β^3)-70 (β^4)+63 (β^5))+α (320-192 β+144 (β^2)-160 (β^3)+140 (β^4)-252 (β^5)+231 (β^6)));
s8=(1/32768)(8960+6435 (α^8)+3432 (α^7) (-2+β)-5120 β+3840 (β^2)-3840 (β^3)+3360 (β^4)-4032 (β^5)+3696 (β^6)-6864 (β^7)+6435 (β^8)+924 (α^6) (4-4 β+3 (β^2))+504 (α^5) (-8+4 β-6 (β^2)+5 (β^3))+70 (α^4) (48-32 β+24 (β^2)-40 (β^3)+35 (β^4))+40 (α^3) (-96+48 β-48 (β^2)+40 (β^3)-70 (β^4)+63 (β^5))+12 (α^2) (320-192 β+144 (β^2)-160 (β^3)+140 (β^4)-252 (β^5)+231 (β^6))+8 α (-640+320 β-288 (β^2)+240 (β^3)-280 (β^4)+252 (β^5)-462 (β^6)+429 (β^7)));
s9=(1/65536)(-17920+12155 (α^9)+6435 (α^8) (-2+β)+8960 β-7680 (β^2)+6400 (β^3)-6720 (β^4)+6048 (β^5)-7392 (β^6)+6864 (β^7)-12870 (β^8)+12155 (β^9)+1716 (α^7) (4-4 β+3 (β^2))+924 (α^6) (-8+4 β-6 (β^2)+5 (β^3))+126 (α^5) (48-32 β+24 (β^2)-40 (β^3)+35 (β^4))+70 (α^4) (-96+48 β-48 (β^2)+40 (β^3)-70 (β^4)+63 (β^5))+20 (α^3) (320-192 β+144 (β^2)-160 (β^3)+140 (β^4)-252 (β^5)+231 (β^6))+12 (α^2) (-640+320 β-288 (β^2)+240 (β^3)-280 (β^4)+252 (β^5)-462 (β^6)+429 (β^7))+α (8960-5120 β+3840 (β^2)-3840 (β^3)+3360 (β^4)-4032 (β^5)+3696 (β^6)-6864 (β^7)+6435 (β^8)));

(Det[(({{s0, s1, s2, s3, s5}, {s1, s2, s3, s4, s6}, {s2, s3, s4, s5, s7}, {s3, s4, s5, s6, s8}, {s4, s5, s6, s7, s9}}))]/(5*Det[(({{s0, s1, s2, s3, s4}, {s1, s2, s3, s4, s5}, {s2, s3, s4, s5, s6}, {s3, s4, s5, s6, s7}, {s4, s5, s6, s7, s8}}))]))//Simplify

(* above is s_n(σ_k) for n = 5, see Formula 23 in [6] *)

Det[(({{s0, s1, s2, s3, s4, s5}, {s1, s2, s3, s4, s5, s6}, {s2, s3, s4, s5, s6, s7}, {s3, s4, s5, s6, s7, s8}, {s4, s5, s6, s7, s8, s9}, {0, s0, s0+s1, s0+s1+s2, s0+s1+s2+s3, s0+s1+s2+s3+s4}}))]//Simplify

(* above is determinant det(M_3) according to Formula 24 in [6], for n = 5 *)