A104746 Array T(n,k) read by antidiagonals: T(1,k) = 2^k-1 and recursively T(n,k) = T(n-1,k) + A000337(k-1), n,k >= 1.

1, 1, 3, 1, 4, 7, 1, 5, 12, 15, 1, 6, 17, 32, 31, 1, 7, 22, 49, 80, 63, 1, 8, 27, 66, 129, 192, 127, 1, 9, 32, 83, 178, 321, 448, 255, 1, 10, 37, 100, 227, 450, 769, 1024, 511, 1, 11, 42, 117, 276, 579, 1090, 1793, 2304, 1023, 1, 12, 47, 134, 325, 708, 1411, 2562, 4097, 5120, 2047, 1, 13, 52, 151, 374, 837, 1732, 3331, 5890, 9217, 11264, 4095

Offset: 1

Author

Gary W. Adamson, Mar 23 2005

Keywords

• nonn
• tabl

Comments

Generally, row n of the array is the binomial transform for 0, 1, n, 2n-1, 3n-2, 4n-3, ...

Examples

			To the first row, add the terms 0, 1, 5, 17, 49, 129, ... as indicated:
  1, 3,  7, 15, 31,  63, ...
  0, 1,  5, 17, 49, 129, ... (getting row 2 of the array:
  1, 4, 12, 32, 80, 192, ... (= A001787, binomial transform for 1,2,3, ...)
Repeat the operation, getting the following array T(n,k):
  1, 3,  7, 15,  31,  63, ...
  1, 4, 12, 32,  80, 192, ...
  1, 5, 17, 49, 129, 321, ...
  1, 6, 22, 66, 178, 450, ...
		

Crossrefs

Cf. A104747 (antidiagonal sums), A001787, A000337, A027992, A059823.

Programs

Maple

A000337 := proc(n)
        1+(n-1)*2^n ;
end proc:
A104746 := proc(n,k)
        option remember;
        if n=  1 then
                2^k-1 ;
        else
                procname(n-1,k)+A000337(k-1) ;
        end if;
end proc:
for d from 1 to 12 do
        for k from 1 to d do
                n := d-k+1 ;
                printf("%d,",A104746(n,k)) ;
        end do:
end do; # R. J. Mathar, Oct 30 2011

Mathematica

A000337[n_] := (n - 1)*2^n + 1;
T[1, k_] := 2^k - 1;
T[n_, k_] := T[n, k] = T[n - 1, k] + A000337[k - 1];
Table[T[n - k + 1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 30 2024 *)

Formula

T(2,k) = A001787(k), binomial transform of 0, 1, 2, 3, 4, 5, 6, ...

T(3,k) = A000337(k), binomial transform of 0, 1, 3, 5, 7, 9, 11, ...

T(4,k) = A027992(k-1), binomial transform of 0, 1, 4, 7, 10, 13, 16, 19, 22, 25, ...

T(5,k) = binomial transform of 0, 1, 5, 9, 13, 17, 21, 25, 29, ...

Extensions

Terms corrected by R. J. Mathar, Oct 30 2011