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%I A055349 #18 Mar 19 2022 16:40:48
%S A055349 1,2,0,6,3,0,24,36,8,0,120,360,220,30,0,720,3600,4200,1500,144,0,5040,
%T A055349 37800,71400,47250,11508,840,0,40320,423360,1176000,1234800,545664,
%U A055349 98784,5760,0,362880,5080320,19474560,29635200,20469456,6618528,940896,45360,0
%N A055349 Triangle of labeled mobiles (circular rooted trees) with n nodes and k leaves.
%H A055349 Andrew Howroyd, <a href="/A055349/b055349.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%H A055349 <a href="/index/Mo#mobiles">Index entries for sequences related to mobiles</a>
%F A055349 E.g.f. satisfies A(x, y) = x*y - x*log(1-A(x, y)). [Corrected by _Sean A. Irvine_, Mar 19 2022]
%e A055349 Triangle begins:
%e A055349      1;
%e A055349      2,     0;
%e A055349      6,     3,     0;
%e A055349     24,    36,     8,     0;
%e A055349    120,   360,   220,    30,     0;
%e A055349    720,  3600,  4200,  1500,   144,   0;
%e A055349   5040, 37800, 71400, 47250, 11508, 840, 0;
%e A055349   ...
%t A055349 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 A055349 T[9] // Flatten (* _Jean-François Alcover_, Oct 31 2019 *)
%o A055349 (PARI)
%o A055349 A(n)={my(v=Vec(serlaplace(serreverse(x/(y - log(1-x + O(x^n))))))); vector(#v, i, Vecrev(v[i]/y, i))}
%o A055349 { my(T=A(10)); for(i=1, #T, print(T[i])) } \\ _Andrew Howroyd_, Sep 23 2018
%Y A055349 Row sums give A038037.
%Y A055349 Columns 1..8 are A000142, A055303, A055350, A055351, A055352, A055353, A055354, A055355.
%Y A055349 Cf. A055340, A055356, A055363.
%K A055349 nonn,tabl,eigen
%O A055349 1,2
%A A055349 _Christian G. Bower_, May 15 2000

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