A244490 Triangle read by rows: T(n,k) (0 <= k <= n) = Sum_{i=0..[k/2]} (-1)^i*binomial(k,2*i)*(2*i-1)!!*n^(k-2*i).

1, 1, 1, 1, 2, 3, 1, 3, 8, 18, 1, 4, 15, 52, 163, 1, 5, 24, 110, 478, 1950, 1, 6, 35, 198, 1083, 5706, 28821, 1, 7, 48, 322, 2110, 13482, 83824, 505876, 1, 8, 63, 488, 3715, 27768, 203569, 1461944, 10270569, 1, 9, 80, 702, 6078, 51894, 436656, 3618540, 29510268, 236644092, 1, 10, 99, 970, 9403, 90150, 854485, 8003950, 74058105, 676549450, 6098971555

Offset: 0

Author

N. J. A. Sloane, Jul 05 2014

Keywords

• nonn
• tabl

Comments

East and Gray (p. 24) give a combinatorial interpretation of the numbers: A function f: Y -> X with Y <= X (<= inclusion) has a 2-cycle if there exists x, y in Y with x != y, f(x) = y and f(y) = x. Then T(n,k) = card({f : [k] -> [n]: f has no 2-cycles}). For instance T(3,3) = 18 because there are 27 functions [3] -> [3], 9 of which have a 2-cycle. - Peter Luschny, Oct 05 2016

Examples

			Triangle begins:
1
1 1
1 2 3
1 3 8 18
1 4 15 52 163
1 5 24 110 478 1950
1 6 35 198 1083 5706 28821
1 7 48 322 2110 13482 83824 505876
1 8 63 488 3715 27768 203569 1461944 10270569
1 9 80 702 6078 51894 436656 3618540 29510268 236644092
...
		

Links

• J. East, R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014.

Crossrefs

As has been noticed by Tom Copeland: T(n,0) = A000012(n), T(n,1) = A001477(n) for n>=1, T(n,2) = A067998(n+1) for n>=3, T(n,3) = A121670(n) for n>=3.

Cf. A276999.

Programs

Maple

T := (n,k) -> add((-1)^i*binomial(k,2*i)*doublefactorial(2*i-1)*n^(k-2*i), i=0..k/2):
seq(seq(T(n,k), k=0..n), n=0..10); # Peter Luschny, Oct 05 2016

Mathematica

Table[Simplify[2^(-k/2) HermiteH[k, n/Sqrt[2]]], {n, 0, 10}, {k,0,n}] // Flatten (* Peter Luschny, Oct 05 2016 *)

Sage

def T(n, k):
    @cached_function
    def h(n, x):
        if n == 0: return 1
        if n == 1: return 2*x
        return 2*(x*h(n-1,x)-(n-1)*h(n-2,x))
    return h(k, n/sqrt(2))/2^(k/2)
for n in range(10):
    print([T(n,k) for k in (0..n)]) # Peter Luschny, Oct 05 2016

Formula

From Peter Luschny, Oct 05 2016: (Start)

T(n,k) = 2^(-k/2)*HermiteH(k, n/sqrt(2)).

T(n,k) = 2^((k-1)/2)*n*KummerU((1-k)/2, 3/2, n^2/2) for n>=1.

T(n,k) = n^k*hypergeom([-k/2, (1-k)/2], [], -2/n^2) for n>=1. (End)