A378622 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the strict partition numbers A000009.

1, 1, 0, 1, 0, 0, 2, 1, 1, 1, 2, 0, -1, -2, -3, 3, 1, 1, 2, 4, 7, 4, 1, 0, -1, -3, -7, -14, 5, 1, 0, 0, 1, 4, 11, 25, 6, 1, 0, 0, 0, -1, -5, -16, -41, 8, 2, 1, 1, 1, 1, 2, 7, 23, 64, 10, 2, 0, -1, -2, -3, -4, -6, -13, -36, -100, 12, 2, 0, 0, 1, 3, 6, 10, 16, 29, 65, 165

Offset

0,7

Links

Table of n, a(n) for n=0..77.

Example

As a table (read by antidiagonals downward):
        n=0:  n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:
  ----------------------------------------------------------
  k=0:   1     1     1     2     2     3     4     5     6
  k=1:   0     0     1     0     1     1     1     1     2
  k=2:   0     1    -1     1     0     0     0     1     0
  k=3:   1    -2     2    -1     0     0     1    -1     0
  k=4:  -3     4    -3     1     0     1    -2     1     1
  k=5:   7    -7     4    -1     1    -3     3     0    -3
  k=6: -14    11    -5     2    -4     6    -3    -3     7
  k=7:  25   -16     7    -6    10    -9     0    10   -14
  k=8: -41    23   -13    16   -19     9    10   -24    24
  k=9:  64   -36    29   -35    28     1   -34    48   -34
As a triangle (read by rows):
   1
   1   0
   1   0   0
   2   1   1   1
   2   0  -1  -2  -3
   3   1   1   2   4   7
   4   1   0  -1  -3  -7 -14
   5   1   0   0   1   4  11  25
   6   1   0   0   0  -1  -5 -16 -41
   8   2   1   1   1   1   2   7  23  64

Mathematica

nn=20;
t=Table[Take[Differences[PartitionsQ/@Range[0, 2nn], k], nn], {k, 0, nn}];
Table[t[[j, i-j+1]], {i, nn/2}, {j, i}]

Crossrefs

Rows are: A000009 (k=0), A087897 (k=1, without first term), A378972 (k=2).

For primes we have A095195 or A376682.

For partitions we have A175804.

First column is A293467 (up to sign).

For composites we have A377033.

For squarefree numbers we have A377038.

For nonsquarefree numbers we have A377046.

For prime powers we have A377051.

Position of first zero in each row is A377285.

Triangle's row-sums are A378970, absolute A378971.

A000009 counts strict integer partitions, differences A087897, A378972.

A000041 counts integer partitions, differences A002865, A053445.

Cf. A047966, A098859, A225486, A325244, A325282.

Cf. A008284, A116608, A325242, A225485 or A325280.

Sequence in context: A088231 A327954 A336386 * A279126 A210679 A143262

Adjacent sequences: A378619 A378620 A378621 * A378625 A378626 A378629

Keyword

sign,tabl,new

Author

Gus Wiseman, Dec 13 2024

Status

approved