

A155861


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


2



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
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OFFSET

0,2


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

Table of n, a(n) for n=0..40.
Almkvist, Gert, "On the differences of the partition function", Acta Arith., 61.2 (1992), 173181.
Knessl, Charles, "Asymptotic Behavior of HighOrder Differences of the Partition Function", Communications on Pure and Applied Mathematics, 44 (1991), 10331045.
Odlyzko, A. M., "Differences of the partition function", Acta Arith., 49 (1988), 237254.
Weisstein, Eric W., "Backward Difference".


FORMULA

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


MAPLE

A41:= n> `if` (n<0, 0, combinat[numbpart](n)):
DB:= proc(p)
proc(n) option remember;
p(n) p(n1)
end
end:
a:= proc(n) option remember;
local f, k;
if n=0 then 1
else f:= (DB@@n)(A41);
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);


CROSSREFS

Cf. A000041, A002865, A053445, A072380, A081094, A081095, A175804, A119712.
Sequence in context: A099416 A211885 A101696 * A212140 A136594 A167826
Adjacent sequences: A155858 A155859 A155860 * A155862 A155863 A155864


KEYWORD

nonn


AUTHOR

Alois P. Heinz, Dec 16 2010


STATUS

approved



