%I #22 Oct 02 2019 10:51:26

%S 1,1,0,1,1,0,1,2,1,0,1,4,3,1,0,1,6,8,4,1,0,1,9,19,16,5,1,0,1,12,37,46,

%T 25,6,1,0,1,16,66,118,96,40,7,1,0,1,20,110,260,300,184,56,8,1,0,1,25,

%U 172,527,811,688,318,80,9,1,0,1,30,257,985,1951,2178,1408,524,105,10,1,0

%N Triangle read by rows: number of mobiles (circular rooted trees) with n nodes and k leaves.

%H Andrew Howroyd, <a href="/A055340/b055340.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)

%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>

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

%F G.f. satisfies A(x, y)=xy+x*CIK(A(x, y))-x. Shifts up under CIK transform.

%F G.f. satisfies A(x, y) = x*(y - Sum_{i>0} phi(i)/i * log(1 - A(x^i, y^i))). - _Michael Somos_, Aug 24 2015

%F Sum_k T(n, k) = A032200(n). - _Michael Somos_, Aug 24 2015

%e G.f. = x^(y + x*y + x^2*(y + y^2) + x^3*(y + 2*y^2 + y^3) + x^4*(y + 4*y^2 + 3*y^3 + y^4) + ...).

%e n\k 1 2 3 4 5 6 7 8

%e --:-- -- -- -- -- -- -- --

%e 1: 1

%e 2: 1 0

%e 3: 1 1 0

%e 4: 1 2 1 0

%e 5: 1 4 3 1 0

%e 6: 1 6 8 4 1 0

%e 7: 1 9 19 16 5 1 0

%e 8: 1 12 37 46 25 6 1 0

%t m = 13; A[_, _] = 0;

%t Do[A[x_, y_] = x (y - Sum[EulerPhi[i]/i Log[1 - A[x^i, y^i]], {i, 1, m}]) + O[x]^m + O[y]^m // Normal, {m}];

%t Join[{1}, Append[CoefficientList[#/y, y], 0]& /@ Rest @ CoefficientList[ A[x, y]/x, x]] // Flatten (* _Jean-François Alcover_, Oct 02 2019 *)

%o (PARI) {T(n, k) = my(A = O(x)); if(k<1 || k>n, 0, for(j=1, n, A = x*y - x*sum(i=1, j, eulerphi(i)/i * log(1 - subst( subst( A + x * O(x^min(j, n\i)), x, x^i), y, y^i) ) )); polcoeff( polcoeff(A, n), k))}; /* _Michael Somos_, Aug 24 2015 */

%Y Row sums give A032200.

%Y Columns 2..8 are A002620(n-1), A055341, A055342, A055343, A055344, A055345, A055346.

%Y Cf. A055347, A055348, A055349, A055356, A055363.

%K nonn,tabl,eigen

%O 1,8

%A _Christian G. Bower_, May 14 2000