A064781 Number of partially labeled rooted trees with n nodes (5 of which are labeled).

625, 7776, 60387, 373895, 2019348, 9941905, 45765687, 200160586, 840783867, 3418136039, 13525307849, 52314181361, 198451152867, 740283848220, 2721325964209, 9875523648986, 35429691205221, 125814678386528, 442690694672091, 1544749563449139, 5349767517169632, 18400000263991476

Offset

5,1

References

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

Links

Table of n, a(n) for n=5..26.

Formula

G.f.: A(x) = B(x)^5*(625-974*B(x)+622*B(x)^2-192*B(x)^3+24*B(x)^4)/(1-B(x))^9, 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]^5*(625 - 974*B[x] + 622*B[x]^2 - 192*B[x]^3 + 24*B[x]^4)/(1 - B[x])^9, {x, 0, terms}], x], 5] (* Vaclav Kotesovec, Jan 10 2026 *)

Crossrefs

Column k=5 of A008295.

Cf. A000081, A042977.

Sequence in context: A046756 A143026 A238700 * A372845 A250832 A250827

Adjacent sequences: A064778 A064779 A064780 * A064782 A064783 A064784

Keyword

nonn

Author

Vladeta Jovovic, Oct 19 2001

Extensions

More terms from Vaclav Kotesovec, Jan 10 2026

Status

approved