A376876 Rademacher's partition formula extended to half-integers. a(n) = round(sqrt(48) * (cosh(h(n)) - sinh(h(n))/h(n)) / (24*n + 11)) where h(n) = sqrt(24*n + 11)*(Pi/6).

1, 1, 2, 4, 6, 9, 13, 18, 26, 36, 49, 66, 88, 117, 154, 202, 262, 338, 435, 555, 705, 891, 1122, 1407, 1757, 2185, 2709, 3347, 4121, 5060, 6194, 7561, 9205, 11178, 13540, 16362, 19727, 23732, 28490, 34132, 40810, 48701, 58011, 68977, 81874, 97019, 114778, 135573

Offset: 0

Author

Peter Luschny, Oct 07 2024

Keywords

• nonn

Comments

One could say that the sequence gives the "number of partitions of n + 1/2", but that would be a linguistic overstretching. However, it does address Richard Stanley's question: "Let n be any complex number in Rademacher's convergent infinite series for p(n) [the number of partitions of n]. For what n does it converge?"

In an answer, Fredrik Johansson points to a cosine-extended version of p(n) that is real-valued on the real line and for which the following also applies: "At half-integers, it appears that all terms in the cosine version of the Rademacher series except the first term vanish, and so one has a trivial closed-form evaluation of p(n + 1/2), n in Z." It is this formula on which the sequence is based.

Examples

			The sequence p(n), {a(n)}, p(n+1), ... (where p(n) = A000041(n)) starts:
1, {1}, 1, {1}, 2, {2}, 3, {4}, 5, {6}, 7, {9}, 11, {13}, 15, {18}, 22, {26}, 30, {36}, 42, {49}, 56, {66}, 77, {88}, 101, ...
In turn, the rounded arithmetic mean q(n) = round((a'(n) + a'(n-1))/2), (where a'(n) is a(n) before rounding) can be seen as a simple approximation to the partition numbers, q(n) ~ p(n). For example p(10^20) = 1.838176508344882643646... *10^11140086259, q(10^20) = 1.838176508344882643649... *10^11140086259.
The sequence q(n) starts: 1, 1, 2, 3, 5, 7, 11, 16, 22, 31, 42, 57, 77, 103, 136, 178, 232, 300, ... For comparison, see the rounded values of the Hardy-Ramanujan approximate formula A050811.
The unrounded value of a(0) is 0.8458241... = A376875.
		

Links

• Fredrik Johansson, Efficient implementation of the Hardy-Ramanujan-Rademacher formula, LMS Journal of Computation and Mathematics, 15:341-59, (2012), MR 2988821, S2CID 16580723.
• Fredrik Johansson, New partition function record: p(10^20) computed, (March 2014).
• Fredrik Johansson, Python program: Rademacher's partition formula, cos case.
• Hans Rademacher, On the partition function p(n), Proceedings of the London Mathematical Society, Second Series, 43 (4):241-254, (1937), MR 1575213.
• Richard Stanley, Does Rademacher's convergent series for p(n) define an analytic function?, and the Answer by Fredrik Johansson, (July 2020).
• Wikipedia, Partition function, approximation formulas.

Crossrefs

Cf. A000041, A050811, A376875.

Programs

Maple

ap := proc(n) local h; h := sqrt(24*n + 11)*(Pi/6):
sqrt(48)*(cosh(h) - sinh(h)/h)/(24*n + 11) end:
seq(round(evalf(ap(n), 64)), n = 0..47);

Mathematica

s[x_] := 1/E^x + E^x + 1/(E^x x) - E^x/x;
h[x_] := Sqrt[11 + 24 x] Pi / 6; g[x_] := Sqrt[12]/(11 + 24 x);
Table[Round[g[n] s[h[n]]], {n, 0, 47}]

Python

# See links.

Formula

a(n) ~ exp((Pi*sqrt(6*n + 3) * (Pi^2*(48*n + 23) - 72) + 6*Pi^2 - 108) / (72*Pi^2*(2*n + 1))) / (sqrt(12)*(2*n + 1)).

Extensions

Dedicated to N. J. A. Sloane on the occasion of his 85th birthday.