cp's OEIS Frontend

A370237 Number of genus 3 partitions of the n-set.

%I A370237 #26 Mar 05 2024 14:49:44
%S A370237 1,94,2620,45430,600655,6633484,64336844,565256120
%N A370237 Number of genus 3 partitions of the n-set.
%C A370237 Call B(n, g) the number of genus g partitions of a set with n elements (genus-dependent Bell number). Then a(n) = B(n, 3) with B(8, 3) = 1.
%C A370237 a(8) = 1 through a(15) = 565256120 were explicitly determined by listing of partitions of an n-set and selecting those of genus 3.
%C A370237 The coefficients of the sixth-degree polynomial appearing in the numerator of the conjectured formula were determined by using experimental values for a(8) up to a(14); the term a(15) given by the formula agrees with the experimental value.
%C A370237 Using the conjectured formula for a(n) gives the following terms for n=16..20 :  4593034160, 35025118700, 253374008888, 1753071498620, 11675101781850. The E.g.f. given in the Formula section is obtained from the conjectured formula for a(n).
%H A370237 Robert Coquereaux and Jean-Bernard Zuber, <a href="https://arxiv.org/abs/2305.01100">Counting partitions by genus. A compendium of results</a>, arXiv:2305.01100 [math.CO], 2023. See p. 8.
%H A370237 Robert Coquereaux and Jean-Bernard Zuber, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Coquereaux/coque5.html">Counting partitions by genus: a compendium of results</a>, Journal of Integer Sequences, Vol. 27 (2024), Article 24.2.6. See p. 9. See also arXiv:2305.01100, 2023.
%F A370237 Conjecture: a(n) = (1/(2^13 * 3^4 * 5 * 7)) * (35*n^6 - 819*n^5 + 7589*n^4 - 36009*n^3 +  93464*n^2 - 129060*n + 95040)/((2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)) * (1/(n-8)!) * (2*n)!/n!.
%F A370237 Conjecture: E.g.f.: (1/181440)*exp(2*x)*(x^2*(720 - 720*x + 1080*x^2 - 720*x^3 + 537*x^4 - 294*x^5 + 140*x^6)*BesselI(0, 2*x) + x*(-720 + 720*x - 1440*x^2 + 1080*x^3 - 1017*x^4 + 594*x^5 - 329*x^6 + 140*x^7)*BesselI(1, 2*x)).
%Y A370237 Column g=3 of A370235.
%Y A370237 Cf. A000108, A002802, A297179, A001263, A370236, A297178.
%K A370237 nonn,more
%O A370237 8,2
%A A370237 _Robert Coquereaux_, Feb 12 2024