A185739 -id:A185739 - cp's OEIS frontend

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

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, 8, 9, 12, 19, 34, 65, 128, 255, 9, 10, 13, 20, 35, 66, 129, 256, 511, 10, 11, 14, 21, 36, 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

Offset: 1

Clark Kimberling, Feb 02 2011

This array fits in a chain: ...->(weight array)->A185738->(accumulation array->...

See the Mathematica code and A144112.

			Northwest corner:
1....2....3....4....5
3....4....5....6....7
7....8....9....10...11
15...16...17...18...19
31...32...33...34...35

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185739, A185740.

(* This program prints the array T=A185738, the accumulation array A185739 of T, and the weight array A185740 of T. *)
f[n_,0]:=0;f[0,k_]:=0;
f[n_,k_]:=2^n+k-2;
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]  (* Array A185738 *)
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
s[n_,k_]:=Sum[f[i,j],{i,1,n},{j,1,k}]; (* accumulation array of {f(n,k)} *)
FullSimplify[s[n,k]]  (* formula for accumulation array *)
TableForm[Table[s[n,k],{n,1,10},{k,1,15}]]  (* Array A185739 *)
Table[s[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten
w[m_,n_]:=f[m,n]+f[m-1,n-1]-f[m,n-1]-f[m-1,n]/;Or[m>0,n>0];
TableForm[Table[w[n,k],{n,1,10},{k,1,15}]] (* Array A185740 *)
Table[w[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

T(n,k) = 2^n + k - 2, n>=1, k>=1.

A185740 Weight array of A185738, by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 0, 4, 1, 0, 0, 8, 1, 0, 0, 0, 16, 1, 0, 0, 0, 0, 32, 1, 0, 0, 0, 0, 0, 64, 1, 0, 0, 0, 0, 0, 0, 128, 1, 0, 0, 0, 0, 0, 0, 0, 256, 1, 0, 0, 0, 0, 0, 0, 0, 0, 512, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1024, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2048, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4096, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8192

Offset: 1

Clark Kimberling, Feb 02 2011

This array is a member of a chain. See A185738. A185740 exemplifies the sort of very simple array whose successive accumulation arrays are interesting. The first two accumulation arrays of A185740 are A185738 and A185739.

			Northwest corner:
1...1...1...1...1...1...1
2...0...0...0...0...0...0
4...0...0...0...0...0...0
8...0...0...0...0...0...0

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A185738.

(See A185738.)
f[n_, k_] := 0; f[n_, 1] := 2^(n - 1); f[1, k_] := 1;
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 10}]] (*Array A185740*)
Table[f[n - k + 1, k], {n, 50}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Jul 12 2017 *)

Column 1: 2^n. Row 1: 1,1,1,1,1,1,1,1,1,1,1,... All other terms: 0.

cp's OEIS Frontend

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

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