%I #28 Dec 15 2015 09:33:35

%S 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,

%T -7,-4,-2,0,4,15,49,28,16,9,5,3,3,7,22,-112,-63,-35,-19,-10,-5,-2,1,8,

%U 30,249,137,74,39,20,10,5,3,4,12,42,-539,-290,-153,-79,-40,-20,-10,-5,-2,2,14,56

%N 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.

%C 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.

%H Alois P. Heinz, <a href="/A175804/b175804.txt">Antidiagonals n = 0..140, flattened</a>

%H Gert Almkvist, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa61/aa6126.pdf">On the differences of the partition function</a>, Acta Arith., 61.2 (1992), 173-181.

%H Charles Knessl, <a href="http://dx.doi.org/10.1002/cpa.3160440814">Asymptotic Behavior of High-Order Differences of the Partition Function</a>, Communications on Pure and Applied Mathematics, 44 (1991), 1033-1045.

%H A. M. Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/doc/arch/partition.fn.diff.pdf">Differences of the partition function</a>, Acta Arith., 49 (1988), 237-254.

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

%e Square array A(n,k) begins:

%e 1, 0, 1, -1, 2, -4, 9, ...

%e 1, 1, 0, 1, -2, 5, -12, ...

%e 2, 1, 1, -1, 3, -7, 16, ...

%e 3, 2, 0, 2, -4, 9, -19, ...

%e 5, 2, 2, -2, 5, -10, 20, ...

%e 7, 4, 0, 3, -5, 10, -20, ...

%e 11, 4, 3, -2, 5, -10, 22, ...

%p A41:= combinat[numbpart]:

%p DD:= proc(p) proc(n) option remember; p(n+1) -p(n) end end:

%p A:= (n,k)-> (DD@@k)(A41)(n):

%p seq(seq(A(n, d-n), n=0..d), d=0..11);

%t 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 *)

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

%Y Cf. A119712, A155861.

%K sign,tabl,look

%O 0,6

%A _Alois P. Heinz_, Dec 04 2010