This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A001901 #54 Feb 16 2025 08:32:24
%S A001901 1,2,4,16,64,128,256,2048,16384,32768,65536,262144,1048576,2097152,
%T A001901 4194304,67108864,1073741824,2147483648,4294967296,17179869184,
%U A001901 68719476736,137438953472,274877906944,2199023255552
%N A001901 Successive numerators of Wallis's approximation to Pi/2 (reduced).
%C A001901 If p is prime, then a(p-2) == - A001902(p-2) (mod p). Cf. A064169 (third comment) and my formula here. Such pseudoprimes are 1467, 7831, ... Primes p such that a(p-2) == - A001902(p-2) (mod p^2) are 5, 45827, ... Cf. A355959, see also A330719 (third comment). - _Thomas Ordowski_, Oct 19 2024
%D A001901 H.-D. Ebbinghaus et al., Numbers, Springer, 1990, p. 146.
%H A001901 Jonathan Sondow, <a href="https://arxiv.org/abs/math/0401406">A faster product for Pi and a new integral for ln(Pi/2)</a>, arXiv:math/0401406 [math.NT], 2004.
%H A001901 Jonathan Sondow, <a href="http://www.jstor.org/stable/30037575">A faster product for Pi and a new integral for ln(Pi/2)</a>, Amer. Math. Monthly 112 (2005), 729-734 and 113 (2006), 670.
%H A001901 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Pi.html">Pi</a>
%H A001901 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PiContinuedFraction.html">Pi Continued Fraction</a>
%H A001901 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F A001901 (2*2*4*4*6*6*8*8*...*2n*2n*...)/(1*3*3*5*5*7*7*9*...*(2n-1)*(2n+1)*...) for n >= 1.
%F A001901 From _Wolfdieter Lang_, Dec 07 2017: (Start)
%F A001901 1/1 * 2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * ...; partial products (reduced). Here the numerators with offset 0.
%F A001901 a(n) = numerator(W(n)), for n >= 0, with W(n) = Product_{k=0..n} N(k)/D(k) (reduced), with N(k) = 2*floor((k+1)/2) for k >= 1 and N(0) = 1, and D(k) = 2*floor(k/2) + 1, for k >= 0. (End)
%F A001901 a(n) is the numerator of the continued fraction [1;1,1/2,1/3,...,1/n]. - _Thomas Ordowski_, Oct 19 2024
%e A001901 From _Wolfdieter Lang_, Dec 07 2017: (Start)
%e A001901 The Wallis numerators (N) and denominators (D) with partial products A(n) = A001900(n) and B(n) = A000246(n+1) in unreduced form, and a(n) and b(n) = A001902(n) in reduced form.
%e A001901 n, k: 0 1 2 3 4 5 6 7 8 9 10 ...
%e A001901 N(k): 1 2 2 4 4 6 6 8 8 10 10 ...
%e A001901 D(k): 1 1 3 3 5 5 7 7 9 9 9 ...
%e A001901 A(n): 1 2 4 16 64 384 2304 18432 147456 1474560 14745600 ...
%e A001901 B(n): 1 1 3 9 45 225 1575 11025 99225 893025 9823275 ...
%e A001901 a(n): 1 2 4 16 64 128 256 2048 16384 32768 65536 ...
%e A001901 b(n): 1 1 3 9 45 75 175 1225 11025 19845 43659 ...
%e A001901 n = 5: numerator(1*2*2*4*4*6/(1*1*3*3*5*5)) = numerator(384/225) = numerator(128/75) = 128. (End)
%t A001901 a[n_?EvenQ] := n!!^2/((n - 1)!!^2*(n + 1)); a[n_?OddQ] := ((n - 1)!!^2*(n + 1))/n!!^2; Table[a[n] // Numerator, {n, 0, 23}] (* _Jean-François Alcover_, Jun 19 2013 *)
%Y A001901 Denominators are A001902. Subsequence of A000079.
%K A001901 nonn,frac,easy
%O A001901 0,2
%A A001901 _N. J. A. Sloane_