A273652 Number of forests of labeled rooted trees of height at most 1, with n labels, two of which are used for root nodes and any root may contain >= 1 labels.

2, 9, 30, 90, 255, 693, 1820, 4644, 11565, 28215, 67650, 159822, 372827, 860265, 1966200, 4456584, 10027161, 22413483, 49807550, 110100690, 242221287, 530579709, 1157628180, 2516582700, 5452595525, 11777605983, 25367150970, 54492397974, 116769423795

Offset

2,1

Links

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Index entries for linear recurrences with constant coefficients, signature (9,-33,63,-66,36,-8).

Formula

E.g.f.: x^2/2! * Sum_{j=0..2} Stirling2(2,j)*exp(x)^j.

a(n) = C(n,2) * Sum_{j=0..2} Stirling2(2,j) * j^(n-2).

G.f.: x^2*(2 - 3*x)*(1 - 3*x + 3*x^2)/((1 - x)*(1 - 2*x))^3. - Andrew Howroyd, Oct 31 2025

Maple

a:= n-> binomial(n, 2)*add(Stirling2(2, j)*j^(n-2), j=0..2):
seq(a(n), n=2..40);

Prog

(PARI) a(n)=((n^2-n)*2^n+4*n^2-4*n)/8 \\ Charles R Greathouse IV, May 31 2026

Crossrefs

Column k=2 of A143396.

Sequence in context: A268586 A056288 A261174 * A056283 A192518 A277241

Adjacent sequences: A273649 A273650 A273651 * A273653 A273654 A273655

Keyword

nonn,easy,changed

Author

Alois P. Heinz, May 27 2016

Status

approved