Let generating function F_d(z) = Sum a_{d,n} z^n generate the counts a_{d,n} of 2*n-step, origin-returning walks on d-dimensional simple-cubic lattice. It is not difficult to prove the following: F_d(z) is the volume integral of a 2*(d-1)-dimensional algebraic variety with symplectic structure, and S_{d-1} permutation symmetry. See below for a the construction in Mathematica, and a few helpful references. --Brad SymplecticGeo[nDim_] := ReplaceAll[Subtract[s^(nDim), Factor@Times[s^(2*nDim), 1 + Total[Plus[ ChebyshevT[2*nDim, X[#]/Sqrt[s]], I ChebyshevU[2*nDim - 1, X[#]/Sqrt[s]] Y[#]/Sqrt[s] ] & /@ Range[nDim]], 1 + Total[Plus[ ChebyshevT[2*nDim, X[#]/Sqrt[s]], -I ChebyshevU[2*nDim - 1, X[#]/Sqrt[s]] Y[#]/Sqrt[s] ] & /@ Range[nDim]]]], s -> (1/nDim) Plus[ Total[X[#]^2 & /@ Range[nDim]], Total[Y[#]^2 & /@ Range[nDim]]]] VolInt[nDim_, Terms_] := TrigReduce[Normal[ Series[Cancel[D[v /. Solve[z == TrigReduce[ SymplecticGeo[nDim] /. { X[n_] :> v^(1/2/nDim) Cos[t[n]], Y[n_] :> v^(1/2/nDim) Sin[t[n]]} ], v][[1]], z]], {z, 0, Terms}]]] /.Cos[_]->0 Column[VolInt[#, 5] & /@ Range[5]] Out[]:= 1 + 4 z + 36 z^2 + 400 z^3 + 4900 z^4 + 63504 z^5 1 + 6 z + 90 z^2 + 1860 z^3 + 44730 z^4 + 1172556 z^5 1 + 8 z + 168 z^2 + 5120 z^3 + 190120 z^4 + 7939008 z^5 1 + 10 z + 270 z^2 + 10900 z^3 + 551950 z^4 + 32232060 z^5 1 + 12 z + 396 z^2 + 19920 z^3 + 1281420 z^4 + 96807312 z^5 Expand[SymplecticGeo[1]] Out[]:= X[1]^2 + Y[1]^2 - 4*X[1]^4 - 4*X[1]^2*Y[1]^2 http://mathworld.wolfram.com/PolyasRandomWalkConstants.html http://demonstrations.wolfram.com/EdwardssSolutionOfPendulumOscillation/ https://oeis.org/A287318 https://oeis.org/A307618