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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.
2
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
1,3
COMMENTS
Generally, row n of the array is the binomial transform for 0, 1, n, 2n-1, 3n-2, 4n-3, ...
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, ...
EXAMPLE
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, ...
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 *)
CROSSREFS
Cf. A104747 (antidiagonal sums), A001787, A000337, A027992, A059823.
Sequence in context: A210218 A086273 A054143 * A350584 A208339 A328463
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Mar 23 2005
EXTENSIONS
Terms corrected by R. J. Mathar, Oct 30 2011
STATUS
approved