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E:=proc(n) local k; if n mod 2 = 0 then k := n/2; if k mod 2 = 0 then RETURN( (k!*(k+2)/2)^2 ); else RETURN( ((k-1)!*(k+1)^2/2)^2 ); fi; else k := (n-1)/2; if k mod 2 = 0 then RETURN( ((k!)^2/12)*(3*k^3+16*k^2+18*k+8) ); else RETURN( ((k-1)!*(k+1)!/12)*(3*k^3+13*k^2-k-3) ); fi; fi; end; # Gives A122749
unprotect(D); D:=proc(n) option remember; if n <= 1 then 1 else D(n-1)+(n-1)*D(n-2); fi; end; # Gives A000085
C:=proc(n) local k; if n mod 2 = 0 then RETURN(0); fi; k:=(n-1)/2; if k mod 2 = 0 then RETURN( k*2^(k-1)*((k/2)!)^2 ); else RETURN( 2^k*(((k+1)/2)!)^2 ); fi; end; # Gives A122693
Q:=proc(n) local m; if n mod 8 <> 1 then RETURN(0); fi; m:=(n-1)/8; ((2*m)!)^2/(m!)^2; end; # Gives A122747
M:=proc(n) local k; if n mod 2 = 0 then k:=n/2; if k mod 2 = 0 then RETURN( k!*(k+2)/2 ); else RETURN( (k-1)!*(k+1)^2/2 ); fi; else k:=(n-1)/2; RETURN(D(k)*D(k+1)); fi; end; # Gives A122748
a:=n-> if n <= 1 then RETURN(1) else E(n)/8 + C(n)/8 + Q(n)/4 + M(n)/4; fi; # Gives A005635
# The following additional Maple programs produce A123071, A005631, A123072, A005633, A005632, A005634
S:=proc(n) local k; if n mod 2 = 0 then RETURN(0) else k:=(n-1)/2; RETURN(B(k)*B(k+1)); fi; end; # Gives A123071
psi:=n->S(n)/2; # Gives A005631
zeta:=n->Q(n)/2; # Gives A123072
mu:=n->(M(n)-S(n))/2; # Gives A005633
chi:=n->(C(n)-S(n)-Q(n))/4; # Gives A005632
eps:=n->E(n)/8-C(n)/8+S(n)/4-M(n)/4; # Gives A005634
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