A276999 - OEIS

%I #22 Mar 14 2020 07:13:21

%S 0,0,0,0,0,1,0,0,1,9,0,0,1,12,93,0,0,1,15,147,1175,0,0,1,18,213,2070,

%T 17835,0,0,1,21,291,3325,33825,317667,0,0,1,24,381,5000,58575,635208,

%U 6506647,0,0,1,27,483,7155,94785,1164429,13536453,150776397

%N Triangle read by rows, T(n,k) = n^k - 2^(k/2)*KummerU(-k/2,1/2,n^2/2) for 0<=k<=n.

%C 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 2-cycles}).

%H J. East, R. D. Gray, <a href="http://arxiv.org/abs/1404.2359">Idempotent generators in finite partition monoids and related semigroups</a>, arXiv preprint arXiv:1404.2359 [math.GR], 2014.

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

%F T(n,k) = n^k - Sum_{i=0..k/2} k!/((-2)^i*i!*(k-2*i)!)*n^(k-2*i).

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

%F T(n,k) ~ n^k*(((k-1)*k)/(2*n^2)-(k*(k^3-6*k^2+11*k-6))/(8*n^4)+(k*(k^5-15*k^4 +85*k^3-225*k^2+274*k-120))/(48*n^6)+O((1/n)^7)).

%e Triangle begins:

%e 0;

%e 0, 0;

%e 0, 0, 1;

%e 0, 0, 1,  9;

%e 0, 0, 1, 12,  93;

%e 0, 0, 1, 15, 147, 1175;

%e 0, 0, 1, 18, 213, 2070, 17835;

%e 0, 0, 1, 21, 291, 3325, 33825,  317667;

%e 0, 0, 1, 24, 381, 5000, 58575,  635208,  6506647;

%e 0, 0, 1, 27, 483, 7155, 94785, 1164429, 13536453, 150776397;

%e .

%e For instance T(3,3) = 9 because there are 27 functions [3]->[3], 18 of which have

%e no 2-cycles. The 9 functions which have 2-cycles are (represented as [f(1), f(2),

%e f(3)]): [1, 3, 2], [2, 1, 1], [2, 1, 2], [2, 1, 3], [2, 3, 2], [3, 1, 1],

%e [3, 2, 1], [3, 3, 1], [3, 3, 2].

%p T := (n,k) -> n^k - 2^(k/2)*KummerU(-k/2, 1/2, n^2/2):

%p seq(seq(simplify(T(n,k)), k=0..n), n=0..9);

%t Table[Simplify[n^k - 2^(-k/2) HermiteH[k, n/Sqrt[2]]], {n, 0, 10}, {k, 0, n}] // Flatten

%o (Sage)

%o def T(n, k):

%o     @cached_function

%o     def h(n, x):

%o         if n == 0: return 1

%o         if n == 1: return 2*x

%o         return 2*(x*h(n-1,x)-(n-1)*h(n-2,x))

%o     return n^k - h(k, n/sqrt(2))/2^(k/2)

%o for n in range(10):

%o     print([T(n,k) for k in (0..n)])

%Y T(n,k) = n^k - A244490(n,k), T(n,3) = A008585(n) for n>=3, T(n,4) = A224334(n-1) for n>=4, T(n,5) = A127694(n-3) for n>=5.

%K nonn,tabl

%O 0,10

%A _Peter Luschny_, Oct 06 2016