A161737 Numerators of the column sums of the BG2 matrix.
1, 2, 16, 128, 2048, 32768, 262144, 2097152, 67108864, 2147483648, 17179869184, 137438953472, 2199023255552, 35184372088832, 281474976710656, 2251799813685248, 144115188075855872, 9223372036854775808, 73786976294838206464, 590295810358705651712, 9444732965739290427392
Offset: 1
Examples
sb(1) = 1; sb(2) = 2; sb(3) = 16/9; sb(4) = 128/75; sb(5) = 2048/1225; etc..
Links
- G. C. Greubel, Table of n, a(n) for n = 1..830
Programs
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Magma
[Numerator((2^(4*n-5)*(Factorial(n-1))^4)/((n-1)*(Factorial(2*n-2))^2)): n in [2..20]]; // G. C. Greubel, Sep 26 2018
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Maple
nmax := 18; x(1):=0: x(2):=1: for n from 2 to nmax-1 do x(n+1) := A050605(n-2) + x(n) + 3 od: for n from 1 to nmax do a(n) := 2^x(n) od: seq(a(n), n=1..nmax); # End program 1 nmax1 := 20; for n from 0 to nmax1 do y(2*n+1) := A090739(n); y(2*n) := A090739(n) od: z(1) := 0: z(2) := 1: for n from 3 to nmax1 do z(n) := z(n-1) + y(n-1) od: for n from 1 to nmax1 do a(n) := 2^z(n) od: seq(a(n), n=1..nmax1); # End program 2 # Above Maple programs edited by Johannes W. Meijer, Sep 25 2012 and by Peter Luschny, Feb 13 2025 r := n -> Pi*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2: a := n -> denom(simplify(r(n))): seq(a(n), n = 1..20); # Peter Luschny, Feb 12 2025
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Mathematica
sb[1] = 1; sb[2] = 2; sb[n_] := sb[n] = sb[n-1]*4*(n-1)*(n-2)/(2n-3)^2; Table[sb[n] // Numerator, {n, 2, 20}] (* Jean-François Alcover, Aug 14 2017 *) -
PARI
vector(20, n, n++; numerator((2^(4*n-5)*(n-1)!^4)/((n-1)*(2*n-2)!^2))) \\ G. C. Greubel, Sep 26 2018
Formula
a(n) = numer(sb(n)) with sb(n) = (2^(4*n-5)*(n-1)!^4)/((n-1)*(2*n-2)!^2) and A161736(n) = denom(sb(n)).
a(n) = denominator(Pi*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2). - Peter Luschny, Feb 12 2025
Extensions
Offset set to 1 and a(1) = 1 prepended by Peter Luschny, Feb 13 2025
Comments