OFFSET
0,6
COMMENTS
Odlyzko showed that the k-th differences of A000041(n) alternate in sign with increasing n up to a certain index n_0(k) and then stay positive.
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
Gert Almkvist, On the differences of the partition function, Acta Arith., 61.2 (1992), 173-181.
Charles Knessl, Asymptotic Behavior of High-Order Differences of the Partition Function, Communications on Pure and Applied Mathematics, 44 (1991), 1033-1045.
A. M. Odlyzko, Differences of the partition function, Acta Arith., 49 (1988), 237-254.
FORMULA
A(n,k) = (Delta^(k) A000041)(n).
EXAMPLE
Square array A(n,k) begins:
1, 0, 1, -1, 2, -4, 9, ...
1, 1, 0, 1, -2, 5, -12, ...
2, 1, 1, -1, 3, -7, 16, ...
3, 2, 0, 2, -4, 9, -19, ...
5, 2, 2, -2, 5, -10, 20, ...
7, 4, 0, 3, -5, 10, -20, ...
11, 4, 3, -2, 5, -10, 22, ...
MAPLE
A41:= combinat[numbpart]:
DD:= proc(p) proc(n) option remember; p(n+1) -p(n) end end:
A:= (n, k)-> (DD@@k)(A41)(n):
seq(seq(A(n, d-n), n=0..d), d=0..11);
MATHEMATICA
max = 11; a41 = Array[PartitionsP, max+1, 0]; a[n_, k_] := Differences[a41, k][[n+1]]; Table[a[n, k-n], {k, 0, max}, {n, 0, k}] // Flatten (* Jean-François Alcover, Aug 29 2014 *)
CROSSREFS
AUTHOR
Alois P. Heinz, Dec 04 2010
STATUS
approved