A155861 -id:A155861 - cp's OEIS frontend

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

Alois P. Heinz, Dec 04 2010

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

			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,  ...

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.

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.

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);

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 *)

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

A119712 a(n) is the smallest integer k such that the n-th (forward) difference of the partition sequence A000041 is positive from k onwards.

Original entry on oeis.org

0, 1, 6, 23, 64, 129, 222, 345, 502, 695, 924, 1193, 1502, 1853, 2246, 2687, 3172, 3705, 4286, 4917, 5600, 6333, 7118, 7957, 8848, 9797, 10800, 11861, 12978, 14153, 15386, 16681, 18034, 19447, 20922, 22459, 24060, 25723, 27448, 29239, 31094, 33015

Offset: 0

Moshe Shmuel Newman, Jun 11 2006

nonn

The first entry is considered to be indexed by zero. For example, the third difference A072380 starts with -1,1 and continues alternating in sign till the 24th entry, from which point it is positive.

Using a different (backward) definition of the difference operator, this sequence has also been given as 1, 8, 26, 68, 134, 228, 352, ... A155861.

Jean-François Alcover, Table of n, a(n) for n = 0..60
Gert Almkvist, On the differences of the partition function, Acta Arith., 61.2 (1992), 173-181.
I. J. Good, Problem 6137, American Mathematical Monthly, 1978, pages 830-831.
Hansraj Gupta, Finite Differences of the Partition Function, Math. Comp. 32 (1978), 1241-1243.
A. M. Odlyzko, Differences of the partition function, Acta Arithmetica 49.3 (1988): 237-254.
Eric Weisstein's World of Mathematics, Forward Difference.

Cf. A000041, A002865, A053445, A072380, A081094, A081095, A175804, A155861.

with(combinat): DD:= proc(p) proc(n) option remember; p(n+1) -p(n) end end: a:= proc(n) option remember; local f, k; if n=0 then 0 else f:= (DD@@n)(numbpart); for k from a(n-1) while not (f(k)>0 and f(k+1)>0) do od; k fi end: seq(a(n), n=0..20); # Alois P. Heinz, Jul 20 2009

a[n_] := a[n] = Module[{f}, f[i_] = DifferenceDelta[PartitionsP[i], {i, n}]; For[j = 2, True, j++, If[f[j] > 0 && f[j+1] > 0, Return[j]]]];
a[0] = 0; a[1] = 1;
Table[Print[n, " ", a[n]]; a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 04 2020 *)

Odlyzko gives an asymptotic formula a(n)~(6/(Pi)^2) * (n log n)^2

a(11)-a(41) from Alois P. Heinz, Jul 20 2009

cp's OEIS Frontend

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.

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