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.

%I A175804 #44 Dec 22 2024 13:29:24
%S A175804 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 A175804 -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 A175804 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 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.
%C A175804 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.
%C A175804 Are there any zeros after the first four, which all lie in columns k = 1, 2? - _Gus Wiseman_, Dec 15 2024
%H A175804 Alois P. Heinz, <a href="/A175804/b175804.txt">Antidiagonals n = 0..140, flattened</a>
%H A175804 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 A175804 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 A175804 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 A175804 A(n,k) = (Delta^(k) A000041)(n).
%F A175804 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
%e A175804 Square array A(n,k) begins:
%e A175804    1,  0,  1, -1,  2,  -4,   9,  ...
%e A175804    1,  1,  0,  1, -2,   5, -12,  ...
%e A175804    2,  1,  1, -1,  3,  -7,  16,  ...
%e A175804    3,  2,  0,  2, -4,   9, -19,  ...
%e A175804    5,  2,  2, -2,  5, -10,  20,  ...
%e A175804    7,  4,  0,  3, -5,  10, -20,  ...
%e A175804   11,  4,  3, -2,  5, -10,  22,  ...
%p A175804 A41:= combinat[numbpart]:
%p A175804 DD:= proc(p) proc(n) option remember; p(n+1) -p(n) end end:
%p A175804 A:= (n,k)-> (DD@@k)(A41)(n):
%p A175804 seq(seq(A(n, d-n), n=0..d), d=0..11);
%t A175804 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 *)
%t A175804 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 *)
%Y A175804 Columns k=0-5 give: A000041, A002865, A053445, A072380, A081094, A081095.
%Y A175804 Main diagonal gives A379378.
%Y A175804 Cf. A119712, A155861.
%Y A175804 For primes we have A095195 or A376682.
%Y A175804 Row n = 0 is A281425.
%Y A175804 Row n = 1 is A320590 except first term.
%Y A175804 For composites we have A377033.
%Y A175804 For squarefree numbers we have A377038.
%Y A175804 For nonsquarefree numbers we have A377046.
%Y A175804 For prime powers we have A377051.
%Y A175804 Antidiagonal sums are A377056, absolute value version A378621.
%Y A175804 The version for strict partitions is A378622, first column A293467.
%Y A175804 A000009 counts strict integer partitions, differences A087897, A378972.
%Y A175804 Cf. A008284, A047966, A098859, A116608, A225486, A325242, A225485 or A325280, A377285, A378970.
%K A175804 sign,tabl,look
%O A175804 0,6
%A A175804 _Alois P. Heinz_, Dec 04 2010