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A185738 Rectangular array T(n,k) = 2^n + k - 2, by antidiagonals.

%I A185738 #10 Jul 12 2017 03:18:30
%S A185738 1,2,3,3,4,7,4,5,8,15,5,6,9,16,31,6,7,10,17,32,63,7,8,11,18,33,64,127,
%T A185738 8,9,12,19,34,65,128,255,9,10,13,20,35,66,129,256,511,10,11,14,21,36,
%U A185738 67,130,257,512,1023,11,12,15,22,37,68,131,258,513,1024,2047,12,13,16,23,38,69,132,259,514,1025,2048,4095,13,14,17,24,39,70,133,260,515,1026,2049,4096,8191,14,15,18,25,40,71,134,261,516,1027,2050,4097,8192,16383
%N A185738 Rectangular array T(n,k) = 2^n + k - 2, by antidiagonals.
%C A185738 This array fits in a chain: ...->(weight array)->A185738->(accumulation array->...
%C A185738 See the Mathematica code and A144112.
%H A185738 G. C. Greubel, <a href="/A185738/b185738.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A185738 T(n,k) = 2^n + k - 2, n>=1, k>=1.
%e A185738 Northwest corner:
%e A185738 1....2....3....4....5
%e A185738 3....4....5....6....7
%e A185738 7....8....9....10...11
%e A185738 15...16...17...18...19
%e A185738 31...32...33...34...35
%t A185738 (* This program prints the array T=A185738, the accumulation array A185739 of T, and the weight array A185740 of T. *)
%t A185738 f[n_,0]:=0;f[0,k_]:=0;
%t A185738 f[n_,k_]:=2^n+k-2;
%t A185738 TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]  (* Array A185738 *)
%t A185738 Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%t A185738 s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
%t A185738 FullSimplify[s[n,k]]  (* formula for accumulation array *)
%t A185738 TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]  (* Array A185739 *)
%t A185738 Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%t A185738 w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0];
%t A185738 TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* Array A185740 *)
%t A185738 Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
%Y A185738 Cf. A144112, A185739, A185740.
%K A185738 nonn,tabl
%O A185738 1,2
%A A185738 _Clark Kimberling_, Feb 02 2011