A003025 Number of n-node labeled acyclic digraphs with 1 out-point. (Formerly M2083)

1, 2, 15, 316, 16885, 2174586, 654313415, 450179768312, 696979588034313, 2398044825254021110, 18151895792052235541515, 299782788128536523836784628, 10727139906233315197412684689421

Offset

1,2

References

R. W. Robinson, Counting labeled acyclic digraphs, pp. 239-273 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Andrew Howroyd, Table of n, a(n) for n = 1..50

Marcel et al., Is there a formula for the number of st-dags (DAG with 1 source and 1 sink) with n vertices?, MathOverflow, 2021.

Example

a(2) = 2: o-->--o (2 ways)
a(3) = 15: o-->--o-->--o (6 ways) and
o ... o o-->--o
.\ . / . \ . /
. v v ... v v
.. o ..... o
(3 ways) (6 ways)

Prog

(PARI) \\ requires A058876.
my(T=A058876(20)); vector(#T, n, T[n][1]) \\ Andrew Howroyd, Dec 27 2021
(PARI) \\ see Marcel et al. link (formula for a').
seq(n)={my(a=vector(n)); a[1]=1; for(n=2, #a, a[n]=sum(k=1, n-1, (-1)^(k-1) * binomial(n, k) * (2^(n-k)-1)^k * a[n-k])); a} \\ Andrew Howroyd, Jan 01 2022

Crossrefs

A diagonal of A058876.

Row sums of A350487.

The unlabeled version is A350415.

Column k=1 of A361718.

Cf. A003026, A003028, A345258.

Sequence in context: A304120 A255929 A059167 * A015200 A331344 A030642

Adjacent sequences: A003022 A003023 A003024 * A003026 A003027 A003028

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Apr 10 2001

Status

approved