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 A278145 #17 Nov 16 2016 03:47:05
%S A278145 1,8,64,1024,16384,131072,1048576,33554432,1073741824,8589934592,
%T A278145 68719476736,1099511627776,17592186044416,140737488355328,
%U A278145 1125899906842624,72057594037927936,4611686018427387904,36893488147419103232,295147905179352825856,4722366482869645213696
%N A278145 Denominator of partial sums of the m=1 member of an m-family of series considered by Hardy with value 4/Pi (see A088538).
%C A278145 The numerators seems to coincide with A161736(n+2).
%C A278145 Hardy considered the m-family of series H(m) = 1/m + (1/(m+1))*(1/2)^2 + (1/(m+2))*(1*3/(2*4))^2 + ... = Sum_{k>=0}(1/(m+k))*(risefac(1/2,k)/k!)^2, where risefac(x,m) = Product_{j=0..m-1} (x+j), and risefac(x,0) = 1. See the Hardy reference, p. 106, eq. (7.5.1) (with n=m).
%C A278145 The value of these series H(m) = (Gamma(m) / Gamma(m+1/2))^2 * Sum_{k = 0..m-1} (risefac(1/2,k)/k!)^2.
%C A278145 The present partial sums are for H(1) with value 1/Gamma(3/2)^2 = 4/Pi (A088538).
%D A278145 G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 106, eq. (7.5.1), and references on p. 112 for Darling (1), p. 232, and Watson (5), p. 235.
%F A278145 a(n)= denominator(r(n)) with the rationals r(n) = Sum_{k=0..n}(1/(k+1))*(risefac(1/2,k)/k!)^2 = Sum_{k=0..n} (1/(k+1))*(binomial(-1/2,k))^2 = Sum_{k=0..n}(1/(k+1))*((2*k-1)!!/(2*k)!!)^2 , with the rising factorial risefac(x,k) defined above. The double factorials are given in A001147 and A000165 with (-1)!! := 1.
%t A278145 Table[Denominator@ Sum[(1/(k + 1)) (Pochhammer[1/2, k]/k!)^2, {k, 0, n}], {n, 0, 19}] (* or *)
%t A278145 Table[Denominator@ Sum[(1/(k + 1)) (Binomial[-1/2, k])^2, {k, 0, n}], {n, 0, 19}] (* or *)
%t A278145 Table[Denominator@ Sum[(1/(k + 1)) ((2 k - 1)!!/(2 k)!!)^2, {k, 0, n}], {n, 0, 19}] (* _Michael De Vlieger_, Nov 15 2016 *)
%Y A278145 Cf. A000165, A001147, A088538, A161736.
%K A278145 nonn,frac,easy
%O A278145 0,2
%A A278145 _Wolfdieter Lang_, Nov 14 2016