%I #31 Oct 25 2018 02:42:28
%S 1,3,13,58,263,1203,5531,25511,117910,545730,2528263,11720917,
%T 54364253,252243996,1170694877,5434421231,25230661483,117153235821,
%U 544024844668,2526465046405,11733602605442,54496414141998,253115681845187,1175659969364675,5460766440081739
%N Number of Schur rings over Z_{5^n}.
%H Gheorghe Coserea, <a href="/A270786/b270786.txt">Table of n, a(n) for n = 0..1001</a>
%H Andrew Misseldine, <a href="http://arxiv.org/abs/1508.03757">Counting Schur Rings over Cyclic Groups</a>, arXiv preprint arXiv:1508.03757 [math.RA], 2015.
%F G.f.: 2*(1-x)/(-2*x^2 + (x-1) + 3*(1-x)*sqrt(1-4*x)); equivalently, the g.f. can be rewritten as -y^2*(y^2 - y + 1)/(2*y^4 - 5*y^3 + 6*y^2 - 5*y + 1), where y=A000108(x). - _Gheorghe Coserea_, Sep 10 2018
%t om[n_] := om[n] = x om[n - 1] + Sum[(CatalanNumber[k - 1] x + 1) om[n - k], {k, 2, n}] // Expand; om[0] = 1; om[1] = x;
%t Array[om, 25, 0] /. x -> 3 (* _Jean-François Alcover_, Oct 25 2018 *)
%o (PARI)
%o A269750_seq(N, t='t) = {
%o my(a=vector(N), c(k)=binomial(2*k, k)/(k+1)); a[1]=1; a[2]=t;
%o for (n = 2, N-1,
%o a[n+1] = t*a[n] + sum(k = 2, n, (c(k-1)*t+1)*a[n+1-k]));
%o return(a);
%o };
%o A269750_seq(25, numdiv(4)) \\ _Gheorghe Coserea_, Sep 10 2018
%Y Cf. A000108, A269750.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Mar 23 2016
%E More terms from _Gheorghe Coserea_, Mar 24 2016
%E a(0)=1 prepended by _Gheorghe Coserea_, Sep 10 2018