cp's OEIS Frontend

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.

%I A155861 #33 Feb 16 2025 08:33:09
%S A155861 1,2,8,26,68,134,228,352,510,704,934,1204,1514,1866,2260,2702,3188,
%T A155861 3722,4304,4936,5620,6354,7140,7980,8872,9822,10826,11888,13006,14182,
%U A155861 15416,16712,18066,19480,20956,22494,24096,25760,27486,29278,31134
%N 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.
%C A155861 Using a different (forward) definition of the difference operator, this sequence has also been given as 0, 1, 6, 23, 64, 129, 222, ... A119712.
%H A155861 Jean-François Alcover, <a href="/A155861/b155861.txt">Table of n, a(n) for n = 0..60</a>
%H A155861 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 A155861 Hansraj Gupta, <a href="https://doi.org/10.1090/S0025-5718-1978-0480319-5">Finite Differences of the Partition Function</a>, Math. Comp. 32 (1978), 1241-1243.
%H A155861 Charles Knessl, <a href="http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160440814/pdf">Asymptotic Behavior of High-Order Differences of the Partition Function</a>, Communications on Pure and Applied Mathematics, 44 (1991), 1033-1045.
%H A155861 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.
%H A155861 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BackwardDifference.html">Backward Difference</a>
%F A155861 An asymptotic formula is a(n) ~ 6/Pi^2 * n^2 (log n)^2.
%p A155861 A41:= n-> `if` (n<0, 0, combinat[numbpart](n)):
%p A155861 DB:= proc(p)
%p A155861        proc(n) option remember;
%p A155861          p(n) -p(n-1)
%p A155861        end
%p A155861      end:
%p A155861 a:= proc(n) option remember;
%p A155861       local f, k;
%p A155861       if n=0 then 1
%p A155861              else f:= (DB@@n)(A41);
%p A155861              for k from a(n-1) while not (f(k)>0 and f(k+1)>0) do od; k
%p A155861       fi
%p A155861     end:
%p A155861 seq(a(n), n=0..20);
%t A155861 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]]]];
%t A155861 a[0] = 1; a[1] = 2;
%t A155861 Table[Print[n, " ", a[n]]; a[n], {n, 0, 60}] (* _Jean-François Alcover_, Dec 04 2020 *)
%Y A155861 Cf. A000041, A002865, A053445, A072380, A081094, A081095, A175804, A119712.
%K A155861 nonn
%O A155861 0,2
%A A155861 _Alois P. Heinz_, Dec 16 2010