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A055363 Triangle of asymmetric mobiles (circular rooted trees) with n nodes and k leaves.

%I A055363 #23 Sep 23 2018 22:32:11
%S A055363 1,1,0,1,0,0,1,1,0,0,1,2,1,0,0,1,4,4,1,0,0,1,6,10,5,1,0,0,1,9,22,19,7,
%T A055363 1,0,0,1,12,42,53,31,8,1,0,0,1,16,73,130,109,45,10,1,0,0,1,20,119,280,
%U A055363 321,190,63,11,1,0,0,1,25,184,556,833,672,310,83,13,1,0,0,1,30,272
%N A055363 Triangle of asymmetric mobiles (circular rooted trees) with n nodes and k leaves.
%H A055363 Andrew Howroyd, <a href="/A055363/b055363.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%H A055363 <a href="/index/Mo#mobiles">Index entries for sequences related to mobiles</a>
%F A055363 G.f. satisfies A(x, y) = x*(y - Sum_{i>0} moebius(i)/i * log(1 - A(x^i, y^i))). - _Michael Somos_, Aug 19 2015
%F A055363 Sum_k T(n, k) = A032171(n). - _Michael Somos_, Aug 24 2015
%e A055363 G.f. = x*(y + x*y + x^2*y + x^3*(y + y^2) + x^4*(y + 2*y^2 + y^3) + x^5*(y + 4*y^2 + 4*y^3 + y^4) + ...).
%e A055363 n\k 1  2  3  4  5  6  7  8
%e A055363 --:-- -- -- -- -- -- -- --
%e A055363 1:  1
%e A055363 2:  1  0
%e A055363 3:  1  0  0
%e A055363 4:  1  1  0  0
%e A055363 5:  1  2  1  0  0
%e A055363 6:  1  4  4  1  0  0
%e A055363 7:  1  6 10  5  1  0  0
%e A055363 8:  1  9 22 19  7  1  0  0
%t A055363 T[n_, k_] := Module[{A}, A[_, _] = 0; If[k<1 || k>n, 0, For[j=1, j <= n, j++, A[x_, y_] = x*y-x*Sum[MoebiusMu[i]/i * Log[1 - A[x^i, y^i]] + O[x]^j // Normal, {i, 1, j}]]; Coefficient[Coefficient[A[x, y], x, n], y, k]]];
%t A055363 Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 30 2017, after _Michael Somos_ *)
%o A055363 (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, moebius(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 A055363 Row sums give A032171.
%Y A055363 Columns 2..8: A002620(n-2), A055364, A055365, A055366, A055367, A055368, A055369.
%Y A055363 Cf. A055340, A055349, A055356, A055370, A055371.
%K A055363 nonn,tabl
%O A055363 1,12
%A A055363 _Christian G. Bower_, May 15 2000