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

Author

Alois P. Heinz, Dec 04 2010

Keywords

• sign
• tabl
• look

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.

Are there any zeros after the first four, which all lie in columns k = 1, 2? - Gus Wiseman, Dec 15 2024

Examples

			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,  ...
		

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.

Crossrefs

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

Main diagonal gives A379378.

Cf. A119712, A155861.

For primes we have A095195 or A376682.

Row n = 0 is A281425.

Row n = 1 is A320590 except first term.

For composites we have A377033.

For squarefree numbers we have A377038.

For nonsquarefree numbers we have A377046.

For prime powers we have A377051.

Antidiagonal sums are A377056, absolute value version A378621.

The version for strict partitions is A378622, first column A293467.

A000009 counts strict integer partitions, differences A087897, A378972.

Cf. A008284, A047966, A098859, A116608, A225486, A325242, A225485 or A325280, A377285, A378970.

Programs

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 *)
nn=5;Table[Table[Sum[(-1)^(k-i)*Binomial[k,i]*PartitionsP[n+i],{i,0,k}],{k,0,nn}],{n,0,nn}] (* Gus Wiseman, Dec 15 2024 *)

Formula

A(n,k) = (Delta^(k) A000041)(n).

A(n,k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k,i) * A000041(n+i). In words, row x is the inverse zero-based binomial transform of A000041 shifted left x times. - Gus Wiseman, Dec 15 2024