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A052144 a(n) = A000172(n)^2.

%I A052144 #19 Feb 02 2024 16:15:24
%S A052144 1,4,100,3136,119716,5071504,230553856,11016601600,546360462244,
%T A052144 27888242788624,1456587070867600,77515424509446400,
%U A052144 4189899499315360000,229472379264509977600,12709952101698593689600,710863065714510068187136
%N A052144 a(n) = A000172(n)^2.
%D A052144 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 191.
%H A052144 Seiichi Manyama, <a href="/A052144/b052144.txt">Table of n, a(n) for n = 0..500</a>
%F A052144 P-recursive: P(n-1)*n^4*a(n) = P(n)*Q(n)*a(n-1) + 8*P(n-1)*Q(n)*a(n-2) - 512*P(n)*(n-2)^4*a(n-3), where P(n) = 7*n^2 - 7*n + 2 and Q(n) = 57*n^4 - 228*n^3 + 321*n^2 - 186*n + 40 with a(0) = 1, a(1) = 4 and a(2) = 100. - _Peter Bala_, Feb 01 2024
%F A052144 a(n) ~ 2^(6*n+2) / (3*Pi^2*n^2). - _Vaclav Kotesovec_, Feb 02 2024
%t A052144 A052144[n_] := HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -1]^2;
%t A052144 Array[A052144, 20, 0] (* _Paolo Xausa_, Jan 30 2024, after _Jean-François Alcover_ in A000172 *)
%Y A052144 Cf. A000172.
%K A052144 nonn,easy
%O A052144 0,2
%A A052144 _N. J. A. Sloane_, Jan 23 2000