A081095 5th differences of partition numbers A000041.

-4, 5, -7, 9, -10, 10, -10, 12, -16, 20, -21, 20, -20, 24, -30, 36, -39, 40, -40, 45, -54, 64, -67, 68, -70, 81, -94, 109, -115, 120, -121, 135, -152, 177, -185, 195, -199, 227, -249, 285, -294, 315, -315, 357, -385, 447, -455, 495, -492, 565, -590, 685, -685, 764, -745, 866, -883, 1047, -1021, 1160

Offset: 0

Author

N. J. A. Sloane, Apr 25 2003

Keywords

• sign

Comments

Comtet appears to say this is nonnegative, which is only true for n sufficiently large.

An explanation is given by Odlyzko. - Moshe Shmuel Newman, Jun 11 2006

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115.
• A. M. Odlyzko, Differences of the partition function, Acta Arith., 49 (1988), pp. 237-254

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
• Almkvist, Gert, "On the differences of the partition function", Acta Arith., 61.2 (1992), 173-181.

Crossrefs

Cf. A000041, A002865, A053445, A072380, A081094.

Programs

Mathematica

Differences[PartitionsP[Range[0,70]],5] (* Harvey P. Dale, Jul 27 2014 *)

Formula

a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^5 / (432 * sqrt(2) * n^(7/2)). - Vaclav Kotesovec, Oct 06 2017