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A156430 Number of n X n arrays of squares of integers, symmetric about both diagonal and antidiagonal, with all rows summing to 2.

%I A156430 #24 Feb 03 2025 21:36:49
%S A156430 1,2,10,12,84,132,954,1728,13290,26820,217500,481320,4086600,9783480,
%T A156430 86549820,221921280,2037627900,5552479800,52745205240,151802154000,
%U A156430 1487961422640,4500041903280,45412066438200,143712079822080,1490217165997560,4917227802767280
%N A156430 Number of n X n arrays of squares of integers, symmetric about both diagonal and antidiagonal, with all rows summing to 2.
%H A156430 Albert Zhou, <a href="/A156430/a156430.pdf">Proof of recurrence</a>
%F A156430 From _Albert Zhou_, Jan 26 2025: (Start)
%F A156430 a(2*n) = a(2*(n-1)) + (n-1)*a(2*(n-2)) + 4*(n-1)*b(2*(n-1)) + 2*(n-1)*(n-2)*c(2*(n-1)), where
%F A156430 b(2*n) = 2*a(2*(n-1)) + 2*(n-1)*b(2*(n-1)), and
%F A156430 c(2*n) = 4*b(2*(n-1)) - 2*a(2*(n-2)) + 4*(n-2)*a(2*(n-3)) + 4*(n-2)*(n-3)*c(2*(n-2)), with
%F A156430 a(0) = 1, a(2) = 1, a(4) = 10, and
%F A156430 b(0) = 0, b(2) = 2, b(4) = 6, and
%F A156430 c(0) = 0, c(2) = 1, c(6) = 6.
%F A156430 a(2*n+1) = n*b(2*n).
%F A156430 Proof attached. (End)
%o A156430 (Python)
%o A156430 # Even-dim bisymmetric
%o A156430 A = [1, 1, 10]
%o A156430 B = [0, 2, 6]
%o A156430 C = [0, 1, 6]
%o A156430 for n in range(3, 13):
%o A156430     a_next = A[-1] + (n-1)*A[-2] + 4*(n-1)*B[-1] + 2*(n-1)*(n-2)*C[-1]
%o A156430     b_next = 2*A[-1] + 2*(n-1)*B[-1]
%o A156430     c_next = 4*B[-1] - 2*A[-2] + 4*(n-2)*A[-3] + 4*(n-2)*(n-3)*C[-2]
%o A156430     A.append(a_next)
%o A156430     B.append(b_next)
%o A156430     C.append(c_next)
%o A156430 # Odd-dim bisymmetric
%o A156430 A_odd = [B[n]*n for n in range(len(B))]
%o A156430 # _Albert Zhou_, Jan 26 2025
%Y A156430 Cf. A001499, A000986.
%K A156430 nonn
%O A156430 2,2
%A A156430 _R. H. Hardin_, Feb 09 2009
%E A156430 a(26)-a(27) from _Albert Zhou_, Jan 26 2025