A175804 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the n-th term of the k-th differences of partition numbers A000041.

1, 0, 1, 1, 1, 2, -1, 0, 1, 3, 2, 1, 1, 2, 5, -4, -2, -1, 0, 2, 7, 9, 5, 3, 2, 2, 4, 11, -21, -12, -7, -4, -2, 0, 4, 15, 49, 28, 16, 9, 5, 3, 3, 7, 22, -112, -63, -35, -19, -10, -5, -2, 1, 8, 30, 249, 137, 74, 39, 20, 10, 5, 3, 4, 12, 42, -539, -290, -153, -79, -40, -20, -10, -5, -2, 2, 14, 56

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

Columns k=0-5 give: A000041, A002865, A053445, A072380, A081094, A081095.

Cf. A119712, A155861.

Sequence in context: A119270 A267109 A341524 * A241063 A340251 A286957

Adjacent sequences: A175801 A175802 A175803 * A175805 A175806 A175807

Keyword

sign,tabl,look

Author

Alois P. Heinz, Dec 04 2010

Status

approved