A105749 Number of ways to use the elements of {1,...,k}, 0 <= k <= 2n, once each to form a sequence of n sets, each having 1 or 2 elements.

1, 2, 14, 222, 6384, 291720, 19445040, 1781750880, 214899027840, 33007837322880, 6290830003852800, 1456812592995513600, 402910665227270323200, 131173228963370155161600, 49656810289225281849907200, 21628258853895305337293568000, 10739534026001485514941587456000

Offset

0,2

Comments

Equivalently, number of sequences of n labeled items such that each item occurs just once or twice. - David Applegate, Dec 08 2008

Also, number of assembly trees for a certain star graph, see Vince-Bona, Theorem 4. - N. J. A. Sloane, Oct 08 2012

Links

Alois P. Heinz, Table of n, a(n) for n = 0..100

R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.

Andrew Vince and Miklos Bona, The Number of Ways to Assemble a Graph, arXiv preprint arXiv:1204.3842 [math.CO], 2012.

Index entries for related partition-counting sequences

Formula

a(n) = Sum_{k=0..n} C(n,k) * (n+k)! / 2^k.

a(n) = n! * A001515(n).

A003011(n) = Sum_{k=0..n} C(n, k)*a(k).

a(n) = Gamma(n+1)*Hypergeometric2F0([-n, n+1], [], -1/2). - Peter Luschny, Jul 29 2014

a(n) ~ sqrt(Pi) * 2^(n + 1) * n^(2*n + 1/2) / exp(2*n - 1). - Vaclav Kotesovec, Nov 27 2017

From G. C. Greubel, Sep 26 2023: (Start)

a(n) = n*(2*n-1)*a(n-1) + n*(n-1)*a(n-2).

a(n) = e * sqrt(2/Pi) * n! * BesselK(n+1/2, 1).

a(n) = ((2*n)!/2^n) * Hypergeometric1F1(-n, -2*n, 2).

G.f.: (-2/x) * Integrate_{t=0..oo} exp(-t)/((t+1)^2 - 1 - 2/x) dt.

G.f.: e*( exp(-sqrt(1 + 2/x)) * ExpIntegralEi(-1 + sqrt(1 + 2/x)) - exp(sqrt(1 + 2/x)) * ExpIntegralEi(-1 - sqrt(1 + 2/x)) )/sqrt(x^2 + 2*x).

E.g.f.: ((1-x)/x) * Hypergeometric1F1(1, 3/2, -(1-x)^2/(2*x)).

E.g.f.: (1/(1-x))*Hypergeometric2F0([1, 1/2]; []; 2*x/(1-x)^2). (End)

Example

a(2) = 14 = |{ ({1},{2}), ({2},{1}), ({1},{2,3}), ({2,3},{1}), ({2},{1,3}), ({1,3},{2}), ({3},{1,2}), ({1,2},{3}), ({1,2},{3,4}), ({3,4},{1,2}), ({1,3},{2,4}), ({2,4},{1,3}), ({1,4},{2,3}), ({2,3},{1,4}) }|.

Maple

a:= n-> add(binomial(n, k)*(n+k)!/2^k, k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 21 2012

Mathematica

f[n_]:= Sum[Binomial[n, k]*(n+k)!/2^k, {k, 0, n}]; Table[f[n], {n, 0, 20}]

Prog

(Magma) [(&+[Binomial(n, j)*Factorial(n+j)/2^j: j in [0..n]]): n in [0..30]]; // G. C. Greubel, Sep 26 2023
(SageMath) [sum(binomial(n, j)*factorial(n+j)//2^j for j in range(n+1)) for n in range(31)] # G. C. Greubel, Sep 26 2023

Crossrefs

Cf. A001515, A003011, A143990.

Replace "sets" by "lists": A099022.

Column n=2 of A181731.

Sequence in context: A372278 A197210 A153668 * A251692 A338187 A323693

Adjacent sequences: A105746 A105747 A105748 * A105750 A105751 A105752

Keyword

nonn,easy

Author

Robert A. Proctor (www.math.unc.edu/Faculty/rap/), Apr 18 2005

Extensions

More terms from Robert G. Wilson v, Apr 23 2005

Status

approved