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%I #25 Aug 29 2019 15:26:00
%S 2,8,1,44,8,2,288,60,24,6,2106,464,228,96,22,16632,3742,2048,1104,440,
%T 91,138996,31392,18246,11328,5940,2184,408,1213056,272592,163896,
%U 111048,68640,33852,11424,1938,10955412,2438208,1493012,1070016,736230,435344,199920,62016,9614,101721744,22369365,13816224,10270752,7602408,5079438,2833152,1209312,346104,49335
%N Triangle read by rows: T(n,k) is the number of rooted maps with n edges whose core comprises k edges, 1 <= k <= n.
%H Gheorghe Coserea, <a href="/A318106/b318106.txt">Rows n=1..202, flattened</a>
%H Cyril Banderier, Philippe Flajolet, Gilles Schaeffer, Michele Soria, <a href="http://algo.inria.fr/flajolet/Publications/BaFlScSo01.pdf">Random maps, coalescing saddles, singularity analysis, and Airy phenomena</a>, Random Structures and Algorithms 19(3-4), 2001.
%F G.f.: A(x;t) = t*h*A000139(t*h), where h=x*A000168(x)^2 (see eqn. (15) in Banderier link).
%e A(x;t) = 2*t*x + (8*t + t^2)*x^2 + (44*t + 8*t^2 + 2*t^3)*x^3 + ...
%e Triangle starts:
%e n\k [1] [2] [3] [4] [5] [6] [7] [8] [9]
%e [1] 2;
%e [2] 8, 1;
%e [3] 44, 8, 2;
%e [4] 288, 60, 24, 6;
%e [5] 2106, 464, 228, 96, 22;
%e [6] 16632, 3742, 2048, 1104, 440, 91;
%e [7] 138996, 31392, 18246, 11328, 5940, 2184, 408;
%e [8] 1213056, 272592, 163896, 111048, 68640, 33852, 11424, 1938;
%e [9] 10955412, 2438208, 1493012, 1070016, 736230, 435344, 199920, 62016, 9614;
%e [10]...
%t A000139[x_] = 2/(3x) (HypergeometricPFQ[{-2/3, -1/3}, {1/2}, (27/4) x]-1);
%t A000168[x_] = HypergeometricPFQ[{1/2, 1}, {3}, 12 x];
%t h[x_] = x A000168[x]^2;
%t A[x_, t_] := t h[x] A000139[t h[x]];
%t Rest[CoefficientList[#, t]]& /@ Rest[CoefficientList[A[x, t] + O[x]^11, x]] // Flatten (* _Jean-François Alcover_, Aug 29 2019 *)
%o (PARI)
%o seq(N) = {
%o my(x='x + O('x^(N+3)), m=(-1 + 18*x + (1-12*x)^(3/2))/(54*x^2),
%o h=x*m^2, c=subst(m, 'x, serreverse(h)));
%o apply(Vecrev, Vec((subst(c, 'x, 't*h) - 1)/'t));
%o };
%o seq(10)
%Y Row sums give A000168 for n>=1.
%Y Main diagonal give A000139(n-1) for n>=1.
%K nonn,tabl
%O 1,1
%A _Gheorghe Coserea_, Sep 22 2018
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