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%I #16 Aug 11 2018 11:34:08
%S 1,2,5,15,49,173,639,2469,9997,43109,205092,1153646,8523086,91156133,
%T 1446766659,32998508358,1047766596136,45632564217917,2711308588849394,
%U 219364550983697100,24151476334929009951,3618445112608409433287
%N Number of unlabeled semitransitive orders on n elements: (1+3)-free posets.
%H M. Guay-Paquet, <a href="http://arxiv.org/abs/1306.2400">A modular relation for the chromatic symmetric functions of (3+1)-free posets</a>, arXiv preprint arXiv:1306.2400 [math.CO], 2013.
%H M. Guay-Paquet, A. H. Morales, E. Rowland, <a href="http://arxiv.org/abs/1212.5356">Structure and enumeration of (3+1)-free posets (extended abstract)</a>, arXiv:1212.5356 [math.CO], 2012.
%F G.f.: S(x/(1-x), T(x)), where S(x, y) is the g.f. for A221494 and T(x) is the g.f. for A221492. [_Mathieu Guay-Paquet_, Jan 18 2013]
%t nmax = 23; co = Coefficient; ex = Exponent;
%t b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i<1, {}, Flatten[Table[Function[ {p}, p + j x^i] /@ b[n - i j, i - 1], {j, 0, n/i}]]]];
%t g[n_, k_] := g[n, k] = Sum[Sum[2^Sum[Sum[GCD[i, j] co[s, x, i] co[t, x, j], {j, 1, ex[t, x]}], {i, 1, ex[s, x]}]/Product[i^co[s, x, i]*co[s, x, i]!, {i, 1, ex[s, x]}]/Product[i^co[t, x, i] co[t, x, i]!, {i, 1, ex[t, x]}], {t, b[n + k, n + k]}], {s, b[n, n]}];
%t A[n_, k_] := g[Min[n, k], Abs[n - k]];
%t A[d_] := Sum[A[n, d - n], {n, 0, d}];
%t B[x_] = Sum[A[n] x^n, {n, 0, nmax}];
%t S[_, _] = 0; Do[S[c_, t_] = Series[1 + (c/(1 + c)) S[c, t]^2 + t S[c, t]^3, {c, 0, nmax}, {t, 0, nmax}] // Normal, {nmax}];
%t T[x_] = 1 - S[x/(1 - x), 1 - 2x - 1/B[x]];
%t Rest[CoefficientList[-T[x] + O[x]^nmax, x]] (* _Jean-François Alcover_, Aug 11 2018, after _Alois P. Heinz_ *)
%Y Cf. A079145 (labeled semitransitive orders), A000112.
%K nonn
%O 1,2
%A Detlef Pauly (dettodet(AT)yahoo.de), Dec 27 2002
%E More terms from _Mathieu Guay-Paquet_, Jan 18 2013
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