OFFSET
0,2
COMMENTS
Row n is the k-th differences of A005117 = the squarefree numbers.
FORMULA
A(i,j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) A005117(i+k).
EXAMPLE
Array form:
n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9:
----------------------------------------------------------
k=0: 1 2 3 5 6 7 10 11 13
k=1: 1 1 2 1 1 3 1 2 1
k=2: 0 1 -1 0 2 -2 1 -1 0
k=3: 1 -2 1 2 -4 3 -2 1 1
k=4: -3 3 1 -6 7 -5 3 0 -2
k=5: 6 -2 -7 13 -12 8 -3 -2 3
k=6: -8 -5 20 -25 20 -11 1 5 -5
k=7: 3 25 -45 45 -31 12 4 -10 10
k=8: 22 -70 90 -76 43 -8 -14 20 -19
k=9: -92 160 -166 119 -51 -6 34 -39 28
Triangle form:
1
2 1
3 1 0
5 2 1 1
6 1 -1 -2 -3
7 1 0 1 3 6
10 3 2 2 1 -2 -8
11 1 -2 -4 -6 -7 -5 3
13 2 1 3 7 13 20 25 22
14 1 -1 -2 -5 -12 -25 -45 -70 -92
15 1 0 1 3 8 20 45 90 160 252
MATHEMATICA
nn=9;
t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1, !SquareFreeQ[#]&]&, 1, 2*nn], k], nn], {k, 0, nn}]
Table[t[[j, i-j+1]], {i, nn}, {j, i}]
CROSSREFS
Row k=0 is A005117.
Row k=1 is A076259.
Row k=2 is A376590.
First position of 0 in each row is A377042.
For nonsquarefree instead of squarefree numbers we have A377046.
For prime-powers instead of squarefree numbers we have A377051.
KEYWORD
AUTHOR
Gus Wiseman, Oct 18 2024
STATUS
approved