%I #18 Mar 19 2022 16:40:48

%S 1,2,0,6,3,0,24,36,8,0,120,360,220,30,0,720,3600,4200,1500,144,0,5040,

%T 37800,71400,47250,11508,840,0,40320,423360,1176000,1234800,545664,

%U 98784,5760,0,362880,5080320,19474560,29635200,20469456,6618528,940896,45360,0

%N Triangle of labeled mobiles (circular rooted trees) with n nodes and k leaves.

%H Andrew Howroyd, <a href="/A055349/b055349.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)

%H <a href="/index/Mo#mobiles">Index entries for sequences related to mobiles</a>

%F E.g.f. satisfies A(x, y) = x*y - x*log(1-A(x, y)). [Corrected by _Sean A. Irvine_, Mar 19 2022]

%e Triangle begins:

%e 1;

%e 2, 0;

%e 6, 3, 0;

%e 24, 36, 8, 0;

%e 120, 360, 220, 30, 0;

%e 720, 3600, 4200, 1500, 144, 0;

%e 5040, 37800, 71400, 47250, 11508, 840, 0;

%e ...

%t T[rows_] := {{1}}~Join~((cc = CoefficientList[#, y]; Append[Rest[cc], 0] * Length[cc]!)& /@ (CoefficientList[InverseSeries[x/(y-Log[1-x + O[x]^rows] ), x], x][[3;;]]));

%t T[9] // Flatten (* _Jean-François Alcover_, Oct 31 2019 *)

%o (PARI)

%o A(n)={my(v=Vec(serlaplace(serreverse(x/(y - log(1-x + O(x^n))))))); vector(#v, i, Vecrev(v[i]/y, i))}

%o { my(T=A(10)); for(i=1, #T, print(T[i])) } \\ _Andrew Howroyd_, Sep 23 2018

%Y Row sums give A038037.

%Y Columns 1..8 are A000142, A055303, A055350, A055351, A055352, A055353, A055354, A055355.

%Y Cf. A055340, A055356, A055363.

%K nonn,tabl,eigen

%O 1,2

%A _Christian G. Bower_, May 15 2000