A318106 - OEIS

%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