1,2

Row 2 of the array = binomial transform for 1, 2, 3...(A001787). Row 3 of the array = binomial transform for 1, 3, 5...(A000337). Row 4 of the array = binomial transform for 1, 4, 7...(A027992). Row 5 of the array = binomial transform for 1, 5, 9...(A059823). Generally, row n of the array = binomial transform for 1, n, (2n - 1)... The operator sequence A000337 (1, 5, 17, 49, 129...) = row 3 of the array, starting with 1. Row sums = A104747: 1, 4, 12, 33, 87, 222, 550...

Construct an array with row 1: 1, 3, 7, 15, 31, 63...; then for the next row, perform the operation, "Add (n-1)th term of A000337 to n-th term of current row, starting with n = 1 of current row". (A000337 = 0, 1, 5, 17, 49, 129, 321...; binomial transform for 1, 3, 5...)

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:

1, 3, 7, 15, 31, 63...

1, 4, 12, 32, 80, 192...

1, 5, 17, 49, 129, 321...

1, 6, 22, 66, 178, 450...

Antidiagonals of the array become rows of the triangle.

Cf. A104747, A001787, A000337, A027992, A059823.

Sequence in context: A081521 A086273 A054143 this_sequence A081255 A005371 A057049

Adjacent sequences: A104743 A104744 A104745 this_sequence A104747 A104748 A104749

nonn,uned

Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 23 2005