A318106 Triangle read by rows: T(n,k) is the number of rooted maps with n edges whose core comprises k edges, 1 <= k <= n.

2, 8, 1, 44, 8, 2, 288, 60, 24, 6, 2106, 464, 228, 96, 22, 16632, 3742, 2048, 1104, 440, 91, 138996, 31392, 18246, 11328, 5940, 2184, 408, 1213056, 272592, 163896, 111048, 68640, 33852, 11424, 1938, 10955412, 2438208, 1493012, 1070016, 736230, 435344, 199920, 62016, 9614, 101721744, 22369365, 13816224, 10270752, 7602408, 5079438, 2833152, 1209312, 346104, 49335

Offset

1,1

Links

Gheorghe Coserea, Rows n=1..202, flattened

Cyril Banderier, Philippe Flajolet, Gilles Schaeffer, Michele Soria, Random maps, coalescing saddles, singularity analysis, and Airy phenomena, Random Structures and Algorithms 19(3-4), 2001.

Formula

G.f.: A(x;t) = t*h*A000139(t*h), where h=x*A000168(x)^2 (see eqn. (15) in Banderier link).

Example

A(x;t) = 2*t*x + (8*t + t^2)*x^2 + (44*t + 8*t^2 + 2*t^3)*x^3 + ...
Triangle starts:
n\k [1]       [2]      [3]      [4]      [5]     [6]     [7]     [8]    [9]
[1] 2;
[2] 8,        1;
[3] 44,       8,       2;
[4] 288,      60,      24,      6;
[5] 2106,     464,     228,     96,      22;
[6] 16632,    3742,    2048,    1104,    440,    91;
[7] 138996,   31392,   18246,   11328,   5940,   2184,   408;
[8] 1213056,  272592,  163896,  111048,  68640,  33852,  11424,  1938;
[9] 10955412, 2438208, 1493012, 1070016, 736230, 435344, 199920, 62016, 9614;
[10]...

Mathematica

A000139[x_] = 2/(3x) (HypergeometricPFQ[{-2/3, -1/3}, {1/2}, (27/4) x]-1);
A000168[x_] = HypergeometricPFQ[{1/2, 1}, {3}, 12 x];
h[x_] = x A000168[x]^2;
A[x_, t_] := t h[x] A000139[t h[x]];
Rest[CoefficientList[#, t]]& /@ Rest[CoefficientList[A[x, t] + O[x]^11, x]] // Flatten (* Jean-François Alcover, Aug 29 2019 *)

Prog

(PARI)
seq(N) = {
  my(x='x + O('x^(N+3)),  m=(-1 + 18*x + (1-12*x)^(3/2))/(54*x^2),
     h=x*m^2, c=subst(m, 'x, serreverse(h)));
  apply(Vecrev, Vec((subst(c, 'x, 't*h) - 1)/'t));
};
seq(10)

Crossrefs

Row sums give A000168 for n>=1.

Main diagonal give A000139(n-1) for n>=1.

Sequence in context: A221074 A065249 A062038 * A308036 A305677 A296569

Adjacent sequences: A318103 A318104 A318105 * A318107 A318108 A318109

Keyword

nonn,tabl

Author

Gheorghe Coserea, Sep 22 2018

Status

approved