A273653 Number of forests of labeled rooted trees of height at most 1, with n labels, three of which are used for root nodes and any root may contain >= 1 labels.

5, 40, 220, 1040, 4550, 19040, 77448, 308640, 1209450, 4668400, 17766892, 66718288, 247397150, 906584000, 3285842960, 11788924992, 41902786770, 147668053200, 516315206260, 1792304871280, 6180666260230, 21184488791840, 72205377800600, 244837696095200

Offset

3,1

Links

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Index entries for linear recurrences with constant coefficients, signature (24,-260,1680,-7206,21600,-46364,71760,-79441,61320,-31320,9504,-1296).

Formula

E.g.f.: x^3/3! * Sum_{j=0..3} Stirling2(3,j)*exp(x)^j.

a(n) = C(n,3) * Sum_{j=0..3} Stirling2(3,j) * j^(n-3).

G.f.: x^3*(5 - 80*x + 560*x^2 - 2240*x^3 + 5620*x^4 - 9120*x^5 + 9428*x^6 - 5712*x^7 + 1555*x^8)/((1 - x)*(1 - 2*x)*(1 - 3*x))^4. - Andrew Howroyd, Oct 31 2025

Maple

a:= n-> binomial(n, 3)*add(Stirling2(3, j)*j^(n-3), j=0..3):
seq(a(n), n=3..40);

Mathematica

A273653[n_] := n*(n-1)*(n-2)*(81*2^n + 8*3^n + 216)/1296;
Array[A273653, 25, 3] (* Paolo Xausa, Jun 17 2026, after Charles R Greathouse IV *)

Prog

(PARI) a(n)=((8*n^3-24*n^2+16*n)*3^n+(81*n^3-243*n^2+162*n)*2^n+216*n^3-648*n^2+432*n)/1296 \\ Charles R Greathouse IV, May 31 2026

Crossrefs

Column k=3 of A143396.

Sequence in context: A272586 A271084 A271092 * A081364 A261739 A230744

Adjacent sequences: A273650 A273651 A273652 * A273654 A273655 A273656

Keyword

nonn,easy,changed

Author

Alois P. Heinz, May 27 2016

Status

approved