OFFSET
0,5
COMMENTS
This simple sequence is such that there is one and only one array of differences D(n,k) where the first and the second upper subdiagonal is a(n).
The rows of this array are existing sequences of the OEIS, prepended with zeros:
row 0 is A118425,
row 1 is A006478,
row 2 is A001629,
row 3 is A010049,
row 4 is A006367,
row 5 is not in the OEIS.
It can be observed that a(n) is an autosequence of the first kind whose second kind mate is A199969. In addition, the structure of the array D(n,k) shows that the first row is an autosequence.
For n = 1 to 8, rows with only one leading zero are also autosequences.
LINKS
OEIS Wiki, Autosequence
Index entries for linear recurrences with constant coefficients, signature (2, -1).
FORMULA
G.f.: x^3 / (1-x)^2.
EXAMPLE
Array of differences begin:
0, 0, 0, 0, 0, 0, 0, 1, 4, 12, 30, 68, ...
0, 0, 0, 0, 0, 0, 1, 3, 8, 18, 38, 76, ...
0, 0, 0, 0, 0, 1, 2, 5, 10, 20, 38, 71, ...
0, 0, 0, 0, 1, 1, 3, 5, 10, 18, 33, 59, ...
0, 0, 0, 1, 0, 2, 2, 5, 8, 15, 26, 46, ...
0, 0, 1, -1, 2, 0, 3, 3, 7, 11, 20, 34, ...
0, 1, -2, 3, -2, 3, 0, 4, 4, 9, 14, 24, ...
1, -3, 5, -5, 5, -3, 4, 0, 5, 5, 10, 16, ...
-4, 8, -10, 10, -8, 7, -4, 5, 0, 6, 6, 17, ...
12, -18, 20, -18, 15, -11, 9, -5, 6, 0, 7, 7, ...
...
MATHEMATICA
a[n_] := Max[0, n - 2];
D[n_, k_] /; k == n + 1 := a[n]; D[n_, k_] /; k == n + 2 := a[n]; D[n_, k_] /; k > n + 2 := D[n, k] = Sum[D[n + 1, j], {j, 0, k - 1}]; D[n_, k_] /; k <= n := D[n, k] = D[n - 1, k + 1] - D[n - 1, k];
Table[D[n, k], {n, 0, 11}, {k, 0, 11}]
CROSSREFS
KEYWORD
nonn,less
AUTHOR
Jean-François Alcover and Paul Curtz, Jun 28 2017
STATUS
approved