OFFSET
0,5
COMMENTS
The same triangle (except for signs) may be obtained from the determinants of the Brahmagupta matrices, setting x -> sqrt(z), y -> 1, t -> n. - Roger L. Bagula, Apr 09 2008
From Bob Selcoe, Nov 15 2014: (Start)
T(n,k)/A000312(n) is the probability P(n,k) that any member (j) of set J={1..n} will be selected k times given n random draws from J. This is equivalent to rolling an n-sided die (with standard assumptions) with sides numbered j=1..n: P(n,k) is the probability that any j will show k times with n rolls.
P(n,k) = (n-2)!*(n-1)^(n-k+1 )/k!*(n-k)!*n^(n-1); n>1. As n approaches infinity, P(n,0) and P(n,1) approach 1/e. (End)
Row sums give n^n (see A000312). - Bob Selcoe, Sep 08 2015
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Eric W. Weisstein's World of Mathematics, Brahmagupta Matrix.
FORMULA
T(n, k) = C(n, k)*(n-1)^(n-k), for n>1.
E.g.f.: (-LambertW(-y)/y)^(x-1)/(1+LambertW(-y)). - Vladeta Jovovic, Feb 19 2004
O.g.f. for row n: (x + n - 1)^n. - Geoffrey Critzer, Mar 21 2010
EXAMPLE
Triangle T(n,k) begins:
1;
0, 1;
1, 2, 1;
8, 12, 6, 1;
81, 108, 54, 12, 1;
1024, 1280, 640, 160, 20, 1;
15625, 18750, 9375, 2500, 375, 30, 1;
279936, 326592, 163296, 45360, 7560, 756, 42, 1;
...
MATHEMATICA
row[n_] := CoefficientList[(x + n - 1)^n + O[x]^(n+1), x];
Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 13 2017, after Geoffrey Critzer *)
Join[{1, 0, 1}, Table[Binomial[n, k]*(n - 1)^(n - k), {n, 2, 49}, {k, 0, n}]] // Flatten (* G. C. Greubel, Nov 14 2017 *)
PROG
(PARI) for(n=0, 9, for(k=0, n, print1(if(n==0 && k==0, 1, if(n==1 && k==0, 0, if(n==1 && k==1, 1, binomial(n, k)*(n-1)^(n-k)))), ", "))) \\ G. C. Greubel, Nov 14 2017
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Christian G. Bower, Apr 25 2000
STATUS
approved