A161381 Triangle read by rows: T(n,k) = n!*2^k/(n-k)! (n >= 0, 0 <= k <= n).

1, 1, 2, 1, 4, 8, 1, 6, 24, 48, 1, 8, 48, 192, 384, 1, 10, 80, 480, 1920, 3840, 1, 12, 120, 960, 5760, 23040, 46080, 1, 14, 168, 1680, 13440, 80640, 322560, 645120, 1, 16, 224, 2688, 26880, 215040, 1290240, 5160960, 10321920, 1, 18, 288, 4032, 48384, 483840, 3870720, 23224320, 92897280, 185794560

Offset: 0

Author

N. J. A. Sloane, Nov 28 2009

Keywords

• nonn
• tabl
• easy

Comments

From Dennis P. Walsh, Nov 20 2012: (Start)

T(n,k) is the number of functions f:[k]->[2n] such that, if f(x)=f(y) or f(x)=2n+1-f(y), then x=y.

We call such functions injective-plus.

Equivalently, T(n,k) gives the number of ways to select k couples from n couples, then choose one person from each of the k selected couples, and then arrange those k individuals in a line. For example, T(50,10) is the number of ways to select 10 U.S. senators, one from each of ten different states, and arrange the senators in a reception line for a visiting dignitary. (End)

Examples

			Triangle begins:
  1
  1  2
  1  4  8
  1  6 24  48
  1  8 48 192  384
  1 10 80 480 1920 3840
For n=2 and k=2, T(2,2)=8 since there are exactly 8 functions f from {1,2} to {1,2,3,4} that are injective-plus. Letting f = <f(1),f(2)>, the 8 functions are <1,2>, <1,3>, <2,1>, <2,4>, <3,1>, <3,4>, <4,2>,and <4,3>. - _Dennis P. Walsh_, Nov 20 2012
		

Links

• Mathieu Guay-Paquet and Jeffrey Shallit, Avoiding Squares and Overlaps Over the Natural Numbers, arXiv:0901.1397 [math.CO], 2009.
• Mathieu Guay-Paquet and Jeffrey Shallit, Avoiding Squares and Overlaps Over the Natural Numbers, Discrete Math., 309 (2009), 6245-6254.
• Dennis Walsh, Notes on injective-plus functions

Crossrefs

A010844 (row sums). Cf. A008279.

Programs

Magma

/* As triangle */ [[Factorial(n)*2^k/Factorial((n-k)): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Dec 23 2015

Maple

seq(seq(2^k*n!/(n-k)!,k=0..n),n=0..20); # Dennis P. Walsh, Nov 20 2012

Mathematica

Flatten@Table[Pochhammer[n - k + 1, k] 2^k, {n, 0, 20}, {k, 0, n}] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 28 2010 *)

Formula

From Dennis P. Walsh, Nov 20 2012: (Start)

E.g.f. for column k: exp(x)*(2*x)^k.

G.f. for column k: (2*x)^k*k!/(1 - x)^(k+1).

T(n,k) = 2^(k-n)*Sum_{j = 0..n} (binomial(n,j)T(j,i)T(n-j,k-i). (End)

From Peter Bala, Feb 20 2016: (Start)

T(n, k) = 2*n*T(n-1, k-1) = 2*k*T(n-1, k-1) + T(n-1, k) = n*T(n-1, k)/(n - k) = 2*(n - k + 1)*T(n, k-1).

G.f. Sum_{n >= 1} (2*n*x*t)^(n-1)/(1 - (2*n*t - 1)*x)^n = 1 + (1 + 2*t)*x + (1 + 4*t + 8*t^2)*x^2 + ....

E.g.f. exp(x)/(1 - 2*x*t) = 1 + (1 + 2*t)*x + (1 + 4*t + 8*t^2)*x^2/2! + ....

E.g.f. for row n: (1 + 2*x)^n.

Row reversed triangle is the exponential Riordan array [1/(1 - 2*x), x]. (End)

Extensions

More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 28 2010