cp's OEIS Frontend

A145656 a(n) = numerator of polynomial of genus 1 and level n for m = 2.

%I A145656 #29 Aug 18 2025 00:42:20
%S A145656 0,2,5,32,131,661,1327,18608,148969,447047,89422,1967410,7869871,
%T A145656 102309709,204620705,2046213056,32739453941,556571077357,556571247527,
%U A145656 10574855234543,42299423848079,42299425233749,84598851790183
%N A145656 a(n) = numerator of polynomial of genus 1 and level n for m = 2.
%C A145656 For the numerators of the polynomials of genus 1 and level n for m = 1 see A001008.
%C A145656 Definition: The polynomial A[1,2*n+1](m) = A[genus 1,level n] is here defined as Sum_{d = 1..n-1} m^(n-d)/d.
%C A145656 First few A[1,n](m):
%C A145656   n = 1: A[1,1](m) = 0
%C A145656   n = 2: A[1,2](m) = m
%C A145656   n = 3: A[1,3](m) = m/2 + m^2
%C A145656   n = 4: A[1,4](m) = m/3 + m^2/2 + m^3
%C A145656   n = 5: A[1,5](m) = m/4 + m^2/3 + m^3/2 + m^4
%C A145656 The general formula which uses these polynomials is the following:
%C A145656 (1/(n+1))*Hypergeometric2F1[1, n, n+1, 1/m] = Sum_{k >= 0} m^(-k)/(k + n) = m^n * ArcTanh[(2*m-1)/(2*m^2-2*m+1)] - A[1,n](m) = (m^n)*Log[m/(m-1)] - A[1,n](m).
%C A145656 Conjecture: a(n) = numerator( (2^n)*log(2) - 2^(n+1)*Integral_{x = 0..1} x^(2*n-1)/(1 + x^2)^n ). - _Peter Bala_, Jun 10 2024
%C A145656 a(n) appears to be a multiple of A068566(n): The sequence {a(n)/A068566(n) : n >= 2} begins [2, 1, 16, 1, 1, 1, 16, 1, 1, 2, 2, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, ...]. - _Peter Bala_, Aug 07 2025
%p A145656 A145656 := proc(n) add( 2^(n-d)/d, d = 1..n-1) end: seq(numer(A145656(n)), n = 1..20); # _R. J. Mathar_, Feb 01 2011
%t A145656 m = 2; aa = {}; Do[k = 0; Do[k = k + m^(r - d)/d, {d, 1, r - 1}]; AppendTo[aa, Numerator[k]], {r, 1, 30}]; aa
%t A145656 (* or *)
%t A145656 a[n_]:=2Integrate[(2-x^n)/(2-x),{x,0,1}]+4(2^(n-1)-1)Log[2]
%t A145656 Table[a[n] // Simplify // Numerator,{n,0,22}] (* _Gerry Martens_, Jun 04 2016 *)
%Y A145656 Cf. A145609-A145640.
%K A145656 frac,nonn,easy
%O A145656 1,2
%A A145656 _Artur Jasinski_, Oct 16 2008