A064782 Number of partially labeled rooted trees with n nodes (6 of which are labeled).

7776, 117649, 1071904, 7601777, 46205469, 252620693, 1277843841, 6088987597, 27669884619, 120970585689, 512143134419, 2110092318654, 8493720340735, 33506301271642, 129860760881905, 495506657571553, 1864611086321279, 6929840762947741, 25467668684988039, 92649817627381902

Offset

6,1

References

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 134.

Links

Table of n, a(n) for n=6..25.

Formula

G.f.: A(x) = B(x)^6*(7776-14543*B(x) +11758*B(x)^2 -5126*B(x)^3 +1200*B(x)^4 -120*B(x)^5)/(1-B(x))^11, where B(x) is g.f. for rooted trees with n nodes, cf. A000081.

Mathematica

terms = 30; B[_] = 0; Do[B[x_] = x*Exp[Sum[B[x^k]/k, {k, 1, j}]] + O[x]^j // Normal, {j, 1, terms}]; Drop[CoefficientList[Series[B[x]^6*(7776 - 14543*B[x] + 11758*B[x]^2 - 5126*B[x]^3 + 1200*B[x]^4 - 120*B[x]^5)/(1 - B[x])^11, {x, 0, terms}], x], 6] (* Vaclav Kotesovec, Jan 10 2026 *)

Crossrefs

Column k=6 of A008295.

Cf. A000081, A042977.

Sequence in context: A386809 A187355 A190465 * A016865 A223561 A016913

Adjacent sequences: A064779 A064780 A064781 * A064783 A064784 A064785

Keyword

nonn

Author

Vladeta Jovovic, Oct 19 2001

Extensions

More terms from Vaclav Kotesovec, Jan 10 2026

Status

approved