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A204207 Triangle based on (1,2,3) averaging array.

%I A204207 #15 Dec 25 2013 13:51:15
%S A204207 2,3,5,5,8,11,9,13,19,23,17,22,32,42,47,33,39,54,74,89,95,65,72,93,
%T A204207 128,163,184,191,129,137,165,221,291,347,375,383,257,266,302,386,512,
%U A204207 638,722,758,767,513,523,568,688,898,1150,1360,1480,1525,1535,1025
%N A204207 Triangle based on (1,2,3) averaging array.
%C A204207 See A204201 for a discussion and guide to other averaging arrays.
%F A204207 T(n,n) = A083329(n). - _Philippe Deléham_, Dec 24 2013
%F A204207 T(n,1) = A000051(n-1). - _Philippe Deléham_, Dec 24 2013
%F A204207 Sum_{k=1..n} T(n,k)=A036289(n). - _Philippe Deléham_, Dec 24 2013
%F A204207 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)=2, T(2,1)=3, T(2,2)=5, T(n,k)=0 if k<1 or if k>n. - _Philippe Deléham_, Dec 24 2013
%e A204207 First six rows:
%e A204207 2
%e A204207 3....5
%e A204207 5....8....11
%e A204207 9....13...19...23
%e A204207 17...22...32...42...47
%e A204207 33...39...54...74...89...95
%t A204207 a = 1; r = 2; b = 3;
%t A204207 t[1, 1] = r;
%t A204207 t[n_, 1] := (a + t[n - 1, 1])/2;
%t A204207 t[n_, n_] := (b + t[n - 1, n - 1])/2;
%t A204207 t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2;
%t A204207 u[n_] := Table[t[n, k], {k, 1, n}]
%t A204207 Table[u[n], {n, 1, 5}]    (* averaging array *)
%t A204207 u = Table[2 (1/2) (1/r) 2^n*u[n], {n, 1, 12}];
%t A204207 TableForm[u] (* A204207 triangle *)
%t A204207 Flatten[u]   (* A204207 sequence *)
%Y A204207 Cf. A204201.
%K A204207 nonn,tabl
%O A204207 1,1
%A A204207 _Clark Kimberling_, Jan 12 2012
%E A204207 Example corrected by _Philippe Deléham_, Dec 22 2013