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

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, 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, 321, 190, 63, 11, 1, 0, 0, 1, 25, 184, 556, 833, 672, 310, 83, 13, 1, 0, 0, 1, 30, 272

Offset: 1

Author

Christian G. Bower, May 15 2000

Keywords

• nonn
• tabl

Examples

			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) + ...).
n\k 1  2  3  4  5  6  7  8
--:-- -- -- -- -- -- -- --
1:  1
2:  1  0
3:  1  0  0
4:  1  1  0  0
5:  1  2  1  0  0
6:  1  4  4  1  0  0
7:  1  6 10  5  1  0  0
8:  1  9 22 19  7  1  0  0
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
• Index entries for sequences related to mobiles

Crossrefs

Row sums give A032171.

Columns 2..8: A002620(n-2), A055364, A055365, A055366, A055367, A055368, A055369.

Cf. A055340, A055349, A055356, A055370, A055371.

Programs

Mathematica

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]]];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 30 2017, after Michael Somos *)

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 */

Formula

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

Sum_k T(n, k) = A032171(n). - Michael Somos, Aug 24 2015