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%I #46 Jun 11 2022 11:43:22
%S 1,1,1,5,8,61,136,1385,3968,50521,176896,2702765,260096,199360981,
%T 951878656,19391512145,104932671488,2404879675441,14544442556416,
%U 74074237647505,2475749026562048,69348874393137901,507711943253426176
%N Denominator of 2*n*A000111(n-1)/A000111(n): approximations of Pi using Euler (up/down) numbers.
%C The rationals r(n)=2*n*e(n-1)/e(n), where e(n)=A000111(n), approximate Pi as n -> oo. - _M. F. Hasler_, Apr 03 2013
%C Numerators are given in A132049.
%C See the Delahaye reference and a link by W. Lang given in A132049.
%C From _Paul Curtz_, Mar 17 2013: (Start)
%C Apply the Akiyama-Tanigawa transform (or algorithm) to A046978(n+2)/A016116(n+1):
%C 1, 1/2, 0, -1/4, -1/4, -1/8, 0, 1/16, 1/16;
%C 1/2, 1, 3/4, 0, -5/8, -3/4, -7/16, 0; = Balmer0(n)
%C -1/2, 1/2, 9/4, 5/2, 5/8, -15/8, -49/16;
%C -1, -7/2, -3/4, 15/2, 25/2, 57/8;
%C 5/2, -11/2, -99/4, -20, 215/8;
%C 8, 77/2, -57/4, -375/2;
%C -61/2, 211/2, 2079/4;
%C -136, -1657/2;
%C 1385/2;
%C The first column is PIEULER(n) = 1, 1/2, -1/2, -1, 5/2, 8, -61/2, -136, 1385/2,... = c(n)/d(n). Abs c(n+1)=1,1,1,5,8,61,... =a(n) with offset=1.
%C For numerators of Balmer0(n) see A076109, A000265 and A061037(n-1) (End).
%C Other completely unrelated rational approximations of Pi are given by A063674/A063673 and other references there. - _M. F. Hasler_, Apr 03 2013
%F a(n)=denominator(r(n)) with the rationals r(n):=2*n*e(n-1)/e(n) where e(n):=A000111(n).
%e Rationals r(n): [2, 4, 3, 16/5, 25/8, 192/61, 427/136, 4352/1385, 12465/3968, 158720/50521, ...].
%t e[n_] := If[EvenQ[n], Abs[EulerE[n]], Abs[(2^(n + 1)*(2^(n + 1) - 1)*BernoulliB[n + 1])/(n + 1)]]; r[n_] := 2*n*(e[n - 1]/e[n]); a[n_] := Denominator[r[n]]; Table[a[n], {n, 1, 23}] (* _Jean-François Alcover_, Mar 26 2013 *)
%o (Python)
%o from itertools import count, islice, accumulate
%o from fractions import Fraction
%o def A132050_gen(): # generator of terms
%o yield 1
%o blist = (0,1)
%o for n in count(2):
%o yield Fraction(2*n*blist[-1],(blist:=tuple(accumulate(reversed(blist),initial=0)))[-1]).denominator
%o A132050_list = list(islice(A132050_gen(),40)) # _Chai Wah Wu_, Jun 09-11 2022
%Y Cf. triangle A008281 (main diagonal give zig-zag numbers A000111).
%K nonn,frac,easy
%O 1,4
%A _Wolfdieter Lang_, Sep 14 2007
%E Definition made more explicit, and initial terms a(1)=a(2)=1 added by _M. F. Hasler_, Apr 03 2013
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