A072380 Third differences of partition numbers A000041.

-1, 1, -1, 2, -2, 3, -2, 3, -2, 5, -4, 7, -3, 7, -3, 11, -5, 15, -4, 17, -2, 24, -4, 32, 1, 38, 5, 53, 7, 70, 18, 86, 33, 115, 45, 152, 74, 191, 109, 254, 150, 331, 218, 420, 307, 551, 410, 716, 567, 913, 767, 1186, 1015, 1529, 1358, 1951, 1799, 2513, 2344, 3222, 3079, 4096, 4009, 5237, 5173

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

Crossrefs

Cf. A000041, A002865, A053445, A081094, A081095.

Programs

Mathematica

Differences[PartitionsP[Range[0,70]],3] (* Harvey P. Dale, Aug 16 2012 *)

Formula

a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^3 / (72 * sqrt(2) * n^(5/2)). - Vaclav Kotesovec, Oct 06 2017