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

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

Author

Clark Kimberling, Jan 12 2012

Keywords

• nonn
• tabl

Comments

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

Examples

			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
		

Crossrefs

Cf. A204202.

Programs

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 *)

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)