(Translated by https://www.hiragana.jp/)
(* Mathematica program for A005635, Jean-Francois Alcover, Jul 23 2022, translated and adapted from N. J. A. Sloane's Maple code *) e[n_]:=Module[{k},If[Mod[n, 2]==0, k=n/2;If[Mod[k, 2]==0, Return[(k!*(k+2)/2)^2], Return[((k-1)!*(k+1)^2/2)^2]], k=(n-1)/2;If[Mod[k, 2]==0, Return[((k!)^2/12)*(3*k^3+16*k^2+18*k+8)], Return[((k-1)!*(k+1)!/12)*(3*k^3+13*k^2-k-3)]]]];(* Gives A122749 *) d[n_]:=d[n]=If[n<=1, 1, d[n-1]+(n-1)*d[n-2]];(* Gives A000085 *) c[n_]:=Module[{k}, If[Mod[n, 2]==0, Return[0]];k=(n-1)/2;If[Mod[k, 2]==0, Return[k*2^(k-1)*((k/2)!)^2], Return[2^k* (((k+1)/2)!)^2]]](* Gives A122693 *) Q[n_]:=Module[{m}, If[Mod[n, 8]!=1, Return[0]]; m=(n-1)/8;((2*m)!)^2/(m!)^2];(* Gives A122747 *) M[n_]:=Module[{k}, If[Mod[n, 2]==0, k=n/2;If[Mod[k, 2]==0, Return[k!*(k+2)/2], Return[(k-1)!*(k+1)^2/2]], k=(n-1)/2; Return[d[k]*d[k+1]]]];(* Gives A122748 *) a[n_]:=If[n<=1, Return[1], e[n]/8+c[n]/8+Q[n]/4+M[n]/4];(* Gives A005635 *) (* The following additional programs produce A123071, A005631, A123072, A005633, A005632, A005634 *) B[n_]:=B[n]=Which[n==0||n==-2, 1, OddQ[n], B[n-1], True, 2*B[n-2]+(n-2)*B[n-4]]; S[n_]:=S[n]=Module[{k}, If[Mod[n, 2]==0, 0, k=(n-1)/2; B[k]*B[k+1]]];(* 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 *) a[n_]:=If[ n<=1, 1, e[n]/8+c[n]/8+Q[n]/4+M[n]/4]; Table[a[n], {n, 0, 30}]