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%I #23 Jul 04 2019 04:50:11
%S 1,3,10,42,35,42,230,42,126,393,230,42,1190,42,230,1158,462,42,3030,
%T 42,1190,1158,230,42,5922,393,230,3030,1190,42,9350,42,1716,1158,230,
%U 1158,20790,42,230,1158,5922,42,9350,42,1190,16782,230,42,28644,393,3030,1158,1190,42,30670,1158,5922,1158,230,42,66290,42,230
%N Number of principal reversible magic squares of order 4n.
%C The reversible magic squares are in a one-to-one correspondence with the most-perfect pandiagonal magic squares (cf. A051235).
%C A reversible magic square composed of integers {0,1,...,n^2-1} is principal if its rows and columns form increasing sequences and the first row starts with 0, 1.
%C Every reversible magic square can be transformed into a unique principal square by (i) interchanging a row/column with the symmetric row/column; and (ii) interchanging two rows/columns in one half of the square and simultaneously interchange their symmetric rows/columns in the other half.
%D K. Ollerenshaw and D. S. Bree, Most-perfect Pan-diagonal Magic Squares: Their Construction and Enumeration, Inst. Math. Applic., Southend-on-Sea, England, 1998.
%D I. Stewart, Most-perfect magic squares, Sci. Amer., Vol. 281, No. 5 (Nov. 1999), pp. 122-123.
%H Max Alekseyev, <a href="/A308951/b308951.txt">Table of n, a(n) for n = 0..10000</a>
%H Steve Abbott, <a href="https://doi.org/10.2307/3619932">Review of Most-perfect Pan-diagonal Magic Squares: Their Construction and Enumeration by Kathleen Ollerenshaw and David Brée</a>, The Mathematical Gazette, Volume 82, Issue 495 November 1998, pp. 535-536.
%F For n>=1, let N := 4n = Product_{g} (p_g)^(s_g), where p_g are distinct primes, and W_v(n) := Sum_{i=0..v} (-1)^(v+i) * binomial(v+1,i+1) * Product_{g} binomial(s_g+i,i). Then a(n) = Sum_{v=0..Sum_{g} s_g} W_v(N)*(W_v(N)+W_{v+1}(N)).
%F For n>=1, a(n) = A051235(n) / 2^(4*n-2) / (2n)!^2.
%t a[n_] := Module[{s, W}, If[n == 0, Return[1]]; s = FactorInteger[4 n][[All, 2]]; W = Table[Sum[(-1)^(V - i - 1) Binomial[V, i + 1] Product[ Binomial[ s[[g]] + i, i], {g, 1, Length[s]}], {i, 0, V - 1}], {V, 1, Total[s] + 1}]; Sum[W[[V]] (W[[V]] + W[[V+1]]), {V, 1, Length[W]-1}]];
%t Table[a[n], {n, 0, 62}] (* _Jean-François Alcover_, Jul 03 2019, translated from PARI *)
%o (PARI) { A308951(n) = if(n==0,return(1)); my(s=factor(4*n)[,2], W=vector(vecsum(s)+1,V, sum(i=0,V-1,(-1)^(V-i-1) * binomial(V,i+1) * prod(g=1,#s,binomial(s[g]+i,i)) ))); sum(V=1,#W-1,W[V]*(W[V]+W[V+1])); }
%Y Cf. A051235.
%K nonn,easy
%O 0,2
%A _Max Alekseyev_, Jul 02 2019
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