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%I #32 Oct 25 2018 02:44:10
%S 1,4,21,113,614,3351,18329,100372,550009,3015021,16531326,90653323,
%T 497159809,2726660260,14954795741,82023673921,449887538266,
%U 2467587944471,13534552583013,74236396347076,407184278878261,2233396783681309,12250146572763594,67191916861119507,368547111683611193
%N Number of Schur rings over Z_{7^n}.
%H Gheorghe Coserea, <a href="/A270787/b270787.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.: (1-x)/(-x^2 + x-1 + 2*(1-x)*sqrt(1-4*x)); equivalently, the g.f. can be rewritten as -y^2*(y^2 - y + 1)/(3*y^4 - 7*y^3 + 8*y^2 - 6*y + 1), where y=A000108(x). - _Gheorghe Coserea_, Sep 10 2018
%t c[k_] := Binomial[2k, k]/(k+1);
%t om[0, _] = 1; om[1, x_] := x; om[n_, x_] := om[n, x] = x om[n-1, x] + Sum[ (c[k-1] x + 1) om[n-k, x], {k, 2, n}];
%t Table[om[n, 4], {n, 0, 25}] (* _Jean-François Alcover_, Oct 05 2018, after _Gheorghe Coserea_ *)
%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(6)) \\ _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
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