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

%I A204201 #12 Dec 24 2013 01:51:13
%S A204201 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,
%T A204201 177,190,1,10,45,122,227,318,367,382,1,11,55,167,349,545,685,749,766,
%U A204201 1,12,66,222,516,894,1230,1434,1515,1534,1,13,78,288,738,1410
%N A204201 Triangle based on (0,1/3,1) averaging array.
%C A204201 For a<r<b, let t(1,1)=r, and for n>1, let
%C A204201 t(n,1)=[a+t(n-1,1)]/2,
%C A204201 t(n,n)=[b+t(n-1,n-1)]/2,
%C A204201 t(n,k)=[t(n-1,k-1)+t(n-1,k)]/2 for 2<=k<=n-1.
%C A204201 We call (t(n,k)) the (a,r,b) averaging array.  If a and b
%C A204201 are integers and r is a rational number, then multiplying
%C A204201 row n of (t(n,k)) by the LCM of its denominators yields a
%C A204201 triangle of integers; A204201 arises in this manner from
%C A204201 (a,r,b)=(0,1/3,1).
%C A204201 ...
%C A204201 Guide to related arrays:
%C A204201 (a,r,b).........triangle
%C A204201 (0,1/2,1).......A054143
%C A204201 (0,1/3,1).......A204201
%C A204201 (0,2/3,1).......A204202
%C A204201 (0,1/4,1).......A204203
%C A204201 (0,3/4,1).......A204204
%C A204201 (0,1/5,1).......A204205
%C A204201 (1,3/2,2).......A204206
%C A204201 (1,2,3).........A204207
%F A204201 From _Philippe Deléham_, Dec 24 2013: (Start)
%F A204201 T(n,n) = A033484(n-1).
%F A204201 Sum{k=1..n} T(n,k) = A053220(n).
%F A204201 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)
%e A204201 The (0,1/3,1) averaging array has these first four rows:
%e A204201 1/3
%e A204201 1/6....2/3
%e A204201 1/12...5/12...5/6
%e A204201 1/24...1/4....5/8...11/12.
%e A204201 Multiplying those rows by 3,6,12,24, respectively:
%e A204201 1
%e A204201 1...4
%e A204201 1...5...10
%e A204201 1...6...15...22
%e A204201 The first nine rows:
%e A204201 1
%e A204201 1...4
%e A204201 1...5...10
%e A204201 1...6...15...22
%e A204201 1...7...21...37...46
%e A204201 1...8...28...58...83...94
%e A204201 1...9...36...86...141..177..190
%e A204201 1...10..45...122..227..318..367..382
%e A204201 1...11..55...167..349..545..685..749..766
%t A204201 a = 0; r = 1/3; b = 1;
%t A204201 t[1, 1] = r;
%t A204201 t[n_, 1] := (a + t[n - 1, 1])/2;
%t A204201 t[n_, n_] := (b + t[n - 1, n - 1])/2;
%t A204201 t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2;
%t A204201 u[n_] := Table[t[n, k], {k, 1, n}]
%t A204201 Table[u[n], {n, 1, 5}]   (* averaging array *)
%t A204201 u = Table[(1/2) (1/r) 2^n*u[n], {n, 1, 12}];
%t A204201 TableForm[u]             (* A204102 triangle *)
%t A204201 Flatten[u]               (* A204201 sequence *)
%Y A204201 Cf. A204202.
%K A204201 nonn,tabl
%O A204201 1,3
%A A204201 _Clark Kimberling_, Jan 12 2012