A391685 Array read by ascending antidiagonals: A(n,k) = (2*n + 1)*2^(2*k+1) - 1 with k >= 0.

1, 5, 7, 9, 23, 31, 13, 39, 95, 127, 17, 55, 159, 383, 511, 21, 71, 223, 639, 1535, 2047, 25, 87, 287, 895, 2559, 6143, 8191, 29, 103, 351, 1151, 3583, 10239, 24575, 32767, 33, 119, 415, 1407, 4607, 14335, 40959, 98303, 131071, 37, 135, 479, 1663, 5631, 18431, 57343, 163839, 393215, 524287

Offset

0,2

Links

Table of n, a(n) for n=0..54.

Formula

G.f.: (1 + x*(3 - 6*y) + 2*y)/((1 - x)^2*(1 - y)*(1 - 4*y)).

E.g.f.: 2*exp(x+4*y)*(1 + 2*x) - exp(x+y).

A(2,n) = (3/2)*A020989(n+1).

A(4,n) = A153465(n+1)/2.

Example

The array begins as:
   1,  7,  31,  127,  511,  2047,  8191, ...
   5, 23,  95,  383, 1535,  6143, 24575, ...
   9, 39, 159,  639, 2559, 10239, 40959, ...
  13, 55, 223,  895, 3583, 14335, 57343, ...
  17, 71, 287, 1151, 4607, 18431, 73727, ...
  21, 87, 351, 1407, 5631, 22527, 90111, ...
  ...

Mathematica

A[n_, k_]:=(2n+1)*2^(2k+1)-1; Table[A[n-k, k], {n, 0, 9}, {k, 0, n}]//Flatten

Crossrefs

Cf. A016813 (column k=0), A083420 (row n=0), A098713 (main diagonal), A140529 (row n=1), A206372 (row n=3), A391686 (antidiagonal sums).

Cf. A020989, A153465.

Sequence in context: A068332 A276734 A268410 * A029650 A049307 A050113

Adjacent sequences: A391682 A391683 A391684 * A391686 A391687 A391688

Keyword

nonn,easy,tabl

Author

Stefano Spezia, Dec 17 2025

Status

approved