A175804 - OEIS

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

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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.

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

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).

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

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 *)

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 *)

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.