A277627 Square array read by antidiagonals downwards: T(n,k), n>=0, k>=0, in which column 0 is equal to A057427: 0, 1, 1, 1, ..., and for k > 0 column k lists two zeros followed by the partial sums of column k-1.

0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 1, 4, 1, 0, 0, 0, 0, 0, 0, 3, 5, 1, 0, 0, 0, 0, 0, 0, 0, 6, 6, 1, 0, 0, 0, 0, 0, 0, 0, 1, 10, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 4, 15, 8, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 21, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 20, 28, 10, 1

Offset: 0

Author

Paul Curtz, Oct 24 2016

Keywords

• nonn
• tabl

Comments

In other words, for n > 0 the column k lists 2*k+1 zeros together with the partial sums of the positive terms of column k-1. - Omar E. Pol, Oct 25 2016

Comments from the author:

1) ZSPEC =

0, 0, 0, 0, 0, 0, 0, 0, ...

1, 0, 0, 0, 0, 0, 0, 0, ...

1, 0, 0, 0, 0, 0, 0, 0, ...

1, 1, 0, 0, 0, 0, 0, 0, ...

1, 2, 0, 0, 0, 0, 0, 0, ...

1, 3, 1, 0, 0, 0, 0, 0, ...

1, 4, 3, 0, 0, 0, 0, 0, ...

1, 5, 6, 1, 0, 0, 0, 0, ...

etc.

The columns are the autosequences of the first kind of the title (column 1: 0, 0, followed by A001477(n); column 2: 0, 0, 0, 0, followed by A000217(n), etc) .

The positive terms are the Pascal triangle written by diagonals (A011973).

First column: A060576(n+1). Or A057427(n), n>-1, thanks to Omar E. Pol.

Row sums: A000045(n), autosequence of the first kind.

Alternated row sums and subtractions: 0, 1, 1, 0, -1, -1, 0 = A128834(n), autosequence of the first kind.

Antidiagonal sums: 0, 1, 1, 1, 2, 3, 4, 6, ... = A078012(n+2).

Application.

Numbers in triangle leading to the Genocchi numbers -A226158(n).

We multiply the columns of ZSPEC by d(n) = 1, -1, 2, -8, 56, -608, ... from A005439.

Hence, with only the first 0,

0,

1,

1,

1, -1,

1, -2,

1, -3, 2,

1, -4, 6,

1, -5, 12, -8,

1, -6, 20, -32,

1, -7, 30, -80, 56,

1, -8, 42, -160, 280,

etc.

The row sums is -A226158(n).

2) Now consider the case of the autosequences of the second kind.

First step.

2, 1, 1, 1, 1, 1, ... = A054977(n)

0, 0, 2, 3, 4, 5, 6, 7, ... = A199969(n) with offset 0

0, 0, 0, 0, 2, 5, 9, 14, 20, 27, ... see A000096

etc.

The positive terms are ASPEC in A191302. By triangle, they are either A029653(n) with A029653(0) = 2 instead of 1 or A029635(n).

Second step. YSPEC =

2, 0, 0, 0, 0, 0, ...

1, 0, 0, 0, 0, 0, ...

1, 2, 0, 0, 0, 0, ...

1, 3, 0, 0, 0, 0, ...

1, 4, 2, 0, 0, 0, ...

1, 5, 5, 0, 0, 0, ...

1, 6, 9, 2, 0, 0, ...

1, 7, 14, 7, 0, 0, ...

etc.

Diagonals by triangle: A029635(n).

This is the companion to ZSPEC.

Row sums: A000032(n), autosequence of the second kind.

Alternated row sums and subtractions: period 6 repeat 2, 1, -1, -2, -1, 1 = A087204(n), autosequence of the second kind.

Application.

Numbers in triangle leading to A230324(n), a companion to -A226158(n).

We multiply the columns of YSPEC by d(n) 1, -1, 2, -8, 56, ... (see above).

Hence, without zeros:

2,

1,

1, -2,

1, -3,

1, -4, 4,

1, -5, 10,

1, -6, 18, -16,

1, -7, 28, -56,

1, -8, 40, -128, 112,

1, -9, 54, -240, 504,

etc.

The row sum is A230324(n).

Links

• G. C. Greubel, Table of n, a(n) for the first 25 antidiagonals, flattened
• OEIS Wiki, Autosequence

Crossrefs

Cf. A011973 (without 0's), A007318 (Pascal's triangle).

Cf. A000045 (row sums), A078012 (antidiagonal sums).

Columns: A060576 or A057427 (k=0), A001477 (k=1), A000217 (k=2).

Cf. A000004, A000012, A000027, A000032, A005439, A029635, A029653, A054977, A087204, A128834, A191302, A199969, A226158, A230324.

Programs

Mathematica

kMax = 13; col[0] = Join[{0}, Array[1&, kMax]]; col[k_] := col[k] = Join[{0, 0}, col[k-1][[1 ;; -3]] // Accumulate]; T[n_, k_] := col[k][[n+1]]; Table[T[n-k, k], {n, 0, kMax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 15 2016 *)

Extensions

Better definition from Omar E. Pol, Oct 25 2016