A376876 - OEIS

%I #29 Dec 19 2024 11:43:26

%S 1,1,2,4,6,9,13,18,26,36,49,66,88,117,154,202,262,338,435,555,705,891,

%T 1122,1407,1757,2185,2709,3347,4121,5060,6194,7561,9205,11178,13540,

%U 16362,19727,23732,28490,34132,40810,48701,58011,68977,81874,97019,114778,135573

%N 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).

%C 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?"

%C 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.

%H Fredrik Johansson, <a href="https://doi.org/10.1112/S1461157012001088">Efficient implementation of the Hardy-Ramanujan-Rademacher formula</a>, LMS Journal of Computation and Mathematics, 15:341-59, (2012), MR 2988821, S2CID 16580723.

%H Fredrik Johansson, <a href="https://fredrikj.net/blog/2014/03/new-partition-function-record/ ">New partition function record: p(10^20) computed</a>, (March 2014).

%H Fredrik Johansson, <a href="/A376876/a376876.pdf">Python program: Rademacher's partition formula, cos case</a>.

%H Hans Rademacher, <a href="https://doi.org/10.1112/plms/s2-43.4.241">On the partition function p(n)</a>, Proceedings of the London Mathematical Society, Second Series, 43 (4):241-254, (1937), MR 1575213.

%H Richard Stanley, <a href="https://mathoverflow.net/q/366733">Does Rademacher's convergent series for p(n) define an analytic function?</a>, and the <a href="https://mathoverflow.net/q/366805">Answer</a> by Fredrik Johansson, (July 2020).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_function_(number_theory)#Approximation_formulas">Partition function, approximation formulas</a>.

%F 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)).

%e The sequence p(n), {a(n)}, p(n+1), ... (where p(n) = A000041(n)) starts:

%e 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, ...

%e 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.

%e 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.

%e The unrounded value of a(0) is 0.8458241... = A376875.

%p ap := proc(n) local h; h := sqrt(24*n + 11)*(Pi/6):

%p sqrt(48)*(cosh(h) - sinh(h)/h)/(24*n + 11) end:

%p seq(round(evalf(ap(n), 64)), n = 0..47);

%t s[x_] := 1/E^x + E^x + 1/(E^x x) - E^x/x;

%t h[x_] := Sqrt[11 + 24 x] Pi / 6; g[x_] := Sqrt[12]/(11 + 24 x);

%t Table[Round[g[n] s[h[n]]], {n, 0, 47}]

%o (Python) # See links.

%Y Cf. A000041, A050811, A376875.

%K nonn

%O 0,3

%A _Peter Luschny_, Oct 07 2024

%E Dedicated to _N. J. A. Sloane_ on the occasion of his 85th birthday.