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A254746 a(n) = Catalan(n^2) mod prime(n).

%I A254746 #12 May 22 2025 10:21:42
%S A254746 1,2,2,5,8,0,0,11,0,24,0,0,29,0,0,0,0,39,58,0,21,33,9,69,9,0,16,86,0,
%T A254746 0,0,0,0,64,139,0,0,0,75,12,4,0,0,119,195,0,193,202,0,0,55,218,0,0,0,
%U A254746 0,84,201,0,0,203,275,0,198,159,0,0,0,0,255,13,204,0
%N A254746 a(n) = Catalan(n^2) mod prime(n).
%H A254746 Chai Wah Wu, <a href="/A254746/b254746.txt">Table of n, a(n) for n = 1..10000</a>
%F A254746 a(n) = A000108(n^2) mod A000040(n).
%t A254746 Table[Mod[CatalanNumber[n^2], Prime[n]], {n, 80}]
%o A254746 (Magma) [Catalan(n^2) mod NthPrime(n): n in [1..100]];
%o A254746 (Python)
%o A254746 from sympy import factorint, prime
%o A254746 A254746_list, c, s, s2 = [1], {}, 2, 4
%o A254746 for n in range(2,10**3+1):
%o A254746     for p,e in factorint(4*n-2).items():
%o A254746         if p in c:
%o A254746             c[p] += e
%o A254746         else:
%o A254746             c[p] = e
%o A254746     for p,e in factorint(n+1).items():
%o A254746         if c[p] == e:
%o A254746             del c[p]
%o A254746         else:
%o A254746             c[p] -= e
%o A254746     if n == s2:
%o A254746         d, ps = 1, prime(s)
%o A254746         for p,e in c.items():
%o A254746             d = (d*pow(p,e,ps)) % ps
%o A254746         A254746_list.append(d)
%o A254746         s2 += 2*s+1
%o A254746         s += 1 # _Chai Wah Wu_, Feb 14 2015
%Y A254746 Cf. A000040, A214441, A246714, A254706.
%K A254746 nonn
%O A254746 1,2
%A A254746 _Vincenzo Librandi_, Feb 07 2015