A204202 -id:A204202 - cp's OEIS frontend

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, 1, 12, 66, 222, 516, 894, 1230, 1434, 1515, 1534, 1, 13, 78, 288, 738, 1410

Offset: 1

Clark Kimberling, Jan 12 2012

For a1, 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

			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

Cf. A204202.

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 *)

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)

cp's OEIS Frontend

A204201 Triangle based on (0,1/3,1) averaging array.

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