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A119712 a(n) is the smallest integer k such that the n-th (forward) difference of the partition sequence A000041 is positive from k onwards. 2
0, 1, 6, 23, 64, 129, 222, 345, 502, 695, 924, 1193, 1502, 1853, 2246, 2687, 3172, 3705, 4286, 4917, 5600, 6333, 7118, 7957, 8848, 9797, 10800, 11861, 12978, 14153, 15386, 16681, 18034, 19447, 20922, 22459, 24060, 25723, 27448, 29239, 31094, 33015 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

Using a different (backward) definition of the difference operator, this sequence has also been given as 1,8,26,68,134,228,352,... A155861.

REFERENCES

I. J. Good, Problem 6137, American Mathematical Monthly 1978 pages 830-831.

Hansraj Gupta, Finite Differences of the Partition Function, pp. 1241-1243.

A. M. Odlyzko, Differences of the partition function, Acta Arith., 49 (1988), pp. 237-254.

LINKS

Table of n, a(n) for n=0..41.

Almkvist, Gert, "On the differences of the partition function", Acta Arith., 61.2 (1992), 173-181.

Weisstein, Eric W. "Forward Difference".

FORMULA

Odlyzko gives an asymptotic formula a(n)~(6/(Pi)^2) * (n log n)^2

MAPLE

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

CROSSREFS

Cf. A000041, A002865, A053445, A072380, A081094, A081095, A175804, A155861.

Sequence in context: A273276 A273252 A208598 * A273314 A281424 A005745

Adjacent sequences:  A119709 A119710 A119711 * A119713 A119714 A119715

KEYWORD

nonn

AUTHOR

Moshe Shmuel Newman, Jun 11 2006

EXTENSIONS

a(11)-a(41) from Alois P. Heinz, Jul 20 2009

STATUS

approved

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Last modified June 23 13:19 EDT 2017. Contains 288665 sequences.