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 A161381 Triangle read by rows: T(n,k) = n!*2^k/(n-k)! (n >= 0, 0 <= k <= n). 0
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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) 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 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) EXAMPLE 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 = , 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 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 *) PROG (MAGMA) /* As triangle */ [[Factorial(n)*2^k/Factorial((n-k)): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Dec 23 2015 CROSSREFS A010844 (row sums). Cf. A008279. Sequence in context: A038557 A011234 A208917 * A220579 A330787 A128412 Adjacent sequences:  A161378 A161379 A161380 * A161382 A161383 A161384 KEYWORD nonn,tabl,easy AUTHOR N. J. A. Sloane, Nov 28 2009 EXTENSIONS More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 28 2010 STATUS approved

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Last modified October 27 02:00 EDT 2021. Contains 348270 sequences. (Running on oeis4.)