OFFSET
1,3
COMMENTS
For a<r<b, let t(1,1)=r, and for n>1, let
t(n,1)=[a+t(n-1,1)]/2,
t(n,n)=[b+t(n-1,n-1)]/2,
t(n,k)=[t(n-1,k-1)+t(n-1,k)]/2 for 2<=k<=n-1.
We call (t(n,k)) the (a,r,b) averaging array. If a and b
are integers and r is a rational number, then multiplying
row n of (t(n,k)) by the LCM of its denominators yields a
triangle of integers; A204201 arises in this manner from
(a,r,b)=(0,1/3,1).
...
Guide to related arrays:
(a,r,b).........triangle
(0,1/2,1).......A054143
(0,1/3,1).......A204201
(0,2/3,1).......A204202
(0,1/4,1).......A204203
(0,3/4,1).......A204204
(0,1/5,1).......A204205
(1,3/2,2).......A204206
(1,2,3).........A204207
FORMULA
From Philippe Deléham, Dec 24 2013: (Start)
T(n,n) = A033484(n-1).
Sum{k=1..n} T(n,k) = A053220(n).
T(n,k) = T(n-1,k)+3*T(n-1,k-1)-2*T(n-2,k-1)-2*T(n-2,k-2), T(1,1)=1, T(2,1)=1, T(2,2)=4, T(n,k)=0 if k<1 or if k>n. (End)
EXAMPLE
The (0,1/3,1) averaging array has these first four rows:
1/3
1/6....2/3
1/12...5/12...5/6
1/24...1/4....5/8...11/12.
Multiplying those rows by 3,6,12,24, respectively:
1
1...4
1...5...10
1...6...15...22
The first nine rows:
1
1...4
1...5...10
1...6...15...22
1...7...21...37...46
1...8...28...58...83...94
1...9...36...86...141..177..190
1...10..45...122..227..318..367..382
1...11..55...167..349..545..685..749..766
MATHEMATICA
a = 0; r = 1/3; b = 1;
t[1, 1] = r;
t[n_, 1] := (a + t[n - 1, 1])/2;
t[n_, n_] := (b + t[n - 1, n - 1])/2;
t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2;
u[n_] := Table[t[n, k], {k, 1, n}]
Table[u[n], {n, 1, 5}] (* averaging array *)
u = Table[(1/2) (1/r) 2^n*u[n], {n, 1, 12}];
TableForm[u] (* A204102 triangle *)
Flatten[u] (* A204201 sequence *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jan 12 2012
STATUS
approved