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A201555 a(n) = C(2*n^2,n^2) = A000984(n^2), where A000984 is the central binomial coefficients.

%I A201555 #40 Jul 09 2022 11:10:30
%S A201555 1,2,70,48620,601080390,126410606437752,442512540276836779204,
%T A201555 25477612258980856902730428600,23951146041928082866135587776380551750,
%U A201555 365907784099042279561985786395502921046971688680,90548514656103281165404177077484163874504589675413336841320
%N A201555 a(n) = C(2*n^2,n^2) = A000984(n^2), where A000984 is the central binomial coefficients.
%C A201555 Central coefficients of triangle A228832.
%H A201555 Vincenzo Librandi, <a href="/A201555/b201555.txt">Table of n, a(n) for n = 0..40</a>
%H A201555 R. Mestrovic, <a href="http://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011)</a>, arXiv:1111.3057 [math.NT], 2011.
%H A201555 R. Oblath, <a href="http://www.ias.ac.in/jarch/proca/1/383-386.pdf">Congruences with binomial coefficients</a>, Proceedings of the Indian Academy of Science, Section A, Vol. 1 No. 6, 383-386
%F A201555 L.g.f.: ignoring initial term, equals the logarithmic derivative of A201556.
%F A201555 a(n) = (2*n^2)! / (n^2)!^2.
%F A201555 a(n) = Sum_{k=0..n^2} binomial(n^2,k)^2.
%F A201555 For primes p >= 5: a(p) == 2 (mod p^3), Oblath, Corollary II; a(p) == binomial(2*p,p) (mod p^6) - see Mestrovic, Section 5, equation 31. - _Peter Bala_, Dec 28 2014
%F A201555 A007814(a(n)) = A159918(n). - _Antti Karttunen_, Apr 27 2017, based on _Vladimir Shevelev_'s Jul 20 2009 formula in A000984.
%e A201555 L.g.f.: L(x) = 2*x + 70*x^2/2 + 48620*x^3/3 + 601080390*x^4/4 + ...
%e A201555 where exponentiation equals the g.f. of A201556:
%e A201555 exp(L(x)) = 1 + 2*x + 37*x^2 + 16278*x^3 + 150303194*x^4 + ... + A201556(n)*x^n + ...
%t A201555 Table[Binomial[2n^2,n^2],{n,0,10}] (* _Harvey P. Dale_, Dec 10 2011 *)
%o A201555 (PARI) a(n) = binomial(2*n^2,n^2)
%o A201555 (Python)
%o A201555 from math import comb
%o A201555 def A201555(n): return comb((m:=n**2)<<1,m) # _Chai Wah Wu_, Jul 08 2022
%Y A201555 Cf. A000984, A007814, A159918, A201556, A214441, A228832, A285388, A285717.
%K A201555 nonn
%O A201555 0,2
%A A201555 _Paul D. Hanna_, Dec 02 2011