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A161737 Numerators of the column sums of the BG2 matrix.

%I A161737 #28 Feb 13 2025 08:26:38
%S A161737 1,2,16,128,2048,32768,262144,2097152,67108864,2147483648,17179869184,
%T A161737 137438953472,2199023255552,35184372088832,281474976710656,
%U A161737 2251799813685248,144115188075855872,9223372036854775808,73786976294838206464,590295810358705651712,9444732965739290427392
%N A161737 Numerators of the column sums of the BG2 matrix.
%C A161737 For the definition of the BG2 matrix coefficients see A161736.
%H A161737 G. C. Greubel, <a href="/A161737/b161737.txt">Table of n, a(n) for n = 1..830</a>
%F A161737 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)).
%F A161737 a(n) = denominator(Pi*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2). - _Peter Luschny_, Feb 12 2025
%e A161737 sb(1) = 1; sb(2) = 2; sb(3) = 16/9; sb(4) = 128/75; sb(5) = 2048/1225; etc..
%p A161737 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
%p A161737 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
%p A161737 # Above Maple programs edited by _Johannes W. Meijer_, Sep 25 2012 and by _Peter Luschny_, Feb 13 2025
%p A161737 r := n -> Pi*(2*n - 2)*((n - 3/2)!/(n - 1)!)^2: a := n -> denom(simplify(r(n))):
%p A161737 seq(a(n), n = 1..20);  # _Peter Luschny_, Feb 12 2025
%t A161737 sb[1] = 1; sb[2] = 2; sb[n_] := sb[n] = sb[n-1]*4*(n-1)*(n-2)/(2n-3)^2;
%t A161737 Table[sb[n] // Numerator, {n, 2, 20}] (* _Jean-François Alcover_, Aug 14 2017 *)
%o A161737 (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
%o A161737 (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
%Y A161737 Cf. A161736 and A161738, A050605, A007814 and A090739.
%K A161737 easy,frac,nonn
%O A161737 1,2
%A A161737 _Johannes W. Meijer_, Jun 18 2009
%E A161737 Offset set to 1 and a(1) = 1 prepended by _Peter Luschny_, Feb 13 2025