OFFSET
0,8
COMMENTS
Comes up in the study of the Zen Stare game (see description at A134362).
T(k,n-k)*binomial(n,k)*(n-k-1)!! is the number of different possible Zen Stare rounds with n starting players and k winners.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325
FORMULA
T(n,k) = Sum_{i=0..n} k^(n-i)*binomial(n,i)*T(i,n-i); This means that with a constant n, T(n,k) is a polynomial of k.
T(n,0) = A134362(n).
T(0,k) = 1.
For odd n, Sum_{k=1..(n+1)/2} T(2*k-1,n-2*k+1)*binomial(n,2*k-1)*(n-2*k)!! = (n-1)^n.
E.g.f. of k-th column: exp((k-1)*W(x) - W(x)^2/2)/(1-W(x)) where W(x) is the e.g.f. of A000169. - Andrew Howroyd, Apr 15 2020
EXAMPLE
Array begins:
=======================================================
n\k | 0 1 2 3 4 5 6
----+--------------------------------------------------
0 | 1 1 1 1 1 1 1 ...
1 | 0 1 2 3 4 5 6 ...
2 | 0 3 8 15 24 35 48 ...
3 | 2 18 52 110 198 322 488 ...
4 | 30 163 478 1083 2110 3715 6078 ...
5 | 444 1950 5706 13482 27768 51894 90150 ...
6 | 7360 28821 83824 203569 436656 854485 1557376 ...
...
T(2,2) = 8; This because given X = {A,B}, Y = {A,B,C,D}. The only functions f: X->Y that meet the requirement are:
f(A) = C, f(B) = C
f(A) = D, f(B) = D
f(A) = D, f(B) = C
f(A) = C, f(B) = D
f(A) = B, f(B) = C
f(A) = B, f(B) = D
f(A) = C, f(B) = A
f(A) = D, f(B) = A
PROG
(PARI) T(n, k)={my(w=-lambertw(-x + O(x^max(4, 1+n)))); n!*polcoef(exp((k-1)*w - w^2/2)/(1-w), n)} \\ Andrew Howroyd, Apr 15 2020
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Mason C. Hart, Apr 14 2020
STATUS
approved