A055340 Triangle read by rows: number of mobiles (circular rooted trees) with n nodes and k leaves.

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

Offset: 1

Author

Christian G. Bower, May 14 2000

Keywords

• nonn
• tabl
• eigen

Examples

			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) + ...).
n\k 1  2  3  4  5  6  7  8
--:-- -- -- -- -- -- -- --
1:  1
2:  1  0
3:  1  1  0
4:  1  2  1  0
5:  1  4  3  1  0
6:  1  6  8  4  1  0
7:  1  9 19 16  5  1  0
8:  1 12 37 46 25  6  1  0
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
• C. G. Bower, Transforms (2)
• Index entries for sequences related to mobiles

Crossrefs

Row sums give A032200.

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

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

Programs

Mathematica

m = 13; A[, ] = 0;
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}];
Join[{1}, Append[CoefficientList[#/y, y], 0]& /@ Rest @ CoefficientList[ A[x, y]/x, x]] // Flatten (* Jean-François Alcover, Oct 02 2019 *)

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 */

Formula

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

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

Sum_k T(n, k) = A032200(n). - Michael Somos, Aug 24 2015