%I
%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 nth (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.
%D I. J. Good, Problem 6137, American Mathematical Monthly 1978 pages 830831.
%D Hansraj Gupta, Finite Differences of the Partition Function, pp. 12411243.
%D A. M. Odlyzko, Differences of the partition function, Acta Arith., 49 (1988), pp. 237254.
%H Almkvist, Gert, "<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), 173181.
%H Weisstein, Eric W. "<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(n1) 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
%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
