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%I #25 Dec 04 2020 14:20:36
%S 0,1,6,23,64,129,222,345,502,695,924,1193,1502,1853,2246,2687,3172,
%T 3705,4286,4917,5600,6333,7118,7957,8848,9797,10800,11861,12978,14153,
%U 15386,16681,18034,19447,20922,22459,24060,25723,27448,29239,31094,33015
%N a(n) is the smallest integer k such that the n-th (forward) difference of the partition sequence A000041 is positive from k onwards.
%C 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.
%C Using a different (backward) definition of the difference operator, this sequence has also been given as 1, 8, 26, 68, 134, 228, 352, ... A155861.
%H Jean-François Alcover, <a href="/A119712/b119712.txt">Table of n, a(n) for n = 0..60</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 I. J. Good, <a href="https://doi.org/10.2307/2320642">Problem 6137</a>, American Mathematical Monthly, 1978, pages 830-831.
%H 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 A. M. Odlyzko, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa49/aa4932.pdf">Differences of the partition function</a>, Acta Arithmetica 49.3 (1988): 237-254.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ForwardDifference.html">Forward Difference</a>.
%F Odlyzko gives an asymptotic formula a(n)~(6/(Pi)^2) * (n log n)^2
%p 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
%t 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]]]];
%t a[0] = 0; a[1] = 1;
%t Table[Print[n, " ", a[n]]; a[n], {n, 0, 60}] (* _Jean-François Alcover_, Dec 04 2020 *)
%Y Cf. A000041, A002865, A053445, A072380, A081094, A081095, A175804, A155861.
%K nonn
%O 0,3
%A _Moshe Shmuel Newman_, Jun 11 2006
%E a(11)-a(41) from _Alois P. Heinz_, Jul 20 2009
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