A155861 a(n) is the smallest integer k such that the n-th (backward) difference of the partition sequence A000041 is positive from k onwards.

1, 2, 8, 26, 68, 134, 228, 352, 510, 704, 934, 1204, 1514, 1866, 2260, 2702, 3188, 3722, 4304, 4936, 5620, 6354, 7140, 7980, 8872, 9822, 10826, 11888, 13006, 14182, 15416, 16712, 18066, 19480, 20956, 22494, 24096, 25760, 27486, 29278, 31134

Offset: 0

Author

Alois P. Heinz, Dec 16 2010

Keywords

• nonn

Comments

Using a different (forward) definition of the difference operator, this sequence has also been given as 0, 1, 6, 23, 64, 129, 222, ... A119712.

Links

• 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.
• Hansraj Gupta, Finite Differences of the Partition Function, Math. Comp. 32 (1978), 1241-1243.
• 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.
• Eric Weisstein's World of Mathematics, Backward Difference

Crossrefs

Cf. A000041, A002865, A053445, A072380, A081094, A081095, A175804, A119712.

Programs

Maple

A41:= n-> `if` (n<0, 0, combinat[numbpart](n)):
DB:= proc(p)
       proc(n) option remember;
         p(n) -p(n-1)
       end
     end:
a:= proc(n) option remember;
      local f, k;
      if n=0 then 1
             else f:= (DB@@n)(A41);
             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);

Mathematica

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 + n]]]];
a[0] = 1; a[1] = 2;
Table[Print[n, " ", a[n]]; a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 04 2020 *)

Formula

An asymptotic formula is a(n) ~ 6/Pi^2 * n^2 (log n)^2.