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A204204 Triangle based on (0,3/4,1) averaging array.

%I A204204 #9 Dec 25 2013 02:29:56
%S A204204 3,3,7,3,10,15,3,13,25,31,3,16,38,56,63,3,19,54,94,119,127,3,22,73,
%T A204204 148,213,246,255,3,25,95,221,361,459,501,511,3,28,120,316,582,820,960,
%U A204204 1012,1023,3,31,148,436,898,1402,1780,1972,2035,2047,3,34,179
%N A204204 Triangle based on (0,3/4,1) averaging array.
%C A204204 See A204201 for a discussion and guide to other averaging arrays.
%F A204204 From _Philippe Deléham_, Dec 24 2013: (Start)
%F A204204 T(n,n) = A000225(n+1).
%F A204204 Sum_{k=1..n} T(n,k) = A128135(n+1).
%F A204204 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)=3, T(2,1)=3, T(2,2)=7, T(n,k)=0 if k<1 or if k>n. (End)
%e A204204 First six rows:
%e A204204 3
%e A204204 3...7
%e A204204 3...10...15
%e A204204 3...13...25...31
%e A204204 3...16...38...56...63
%e A204204 3...19...54...94...119..127
%t A204204 a = 0; r = 3/4; b = 1;
%t A204204 t[1, 1] = r;
%t A204204 t[n_, 1] := (a + t[n - 1, 1])/2;
%t A204204 t[n_, n_] := (b + t[n - 1, n - 1])/2;
%t A204204 t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2;
%t A204204 u[n_] := Table[t[n, k], {k, 1, n}]
%t A204204 Table[u[n], {n, 1, 5}]   (* averaging array *)
%t A204204 u = Table[3 (1/2) (1/r) 2^n*u[n], {n, 1, 12}];
%t A204204 TableForm[u]  (* A204204 triangle *)
%t A204204 Flatten[u]    (* A204204 sequence *)
%Y A204204 Cf. A204201.
%K A204204 nonn,tabl
%O A204204 1,1
%A A204204 _Clark Kimberling_, Jan 12 2012