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 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). 3
 1, 8, 64, 1024, 16384, 131072, 1048576, 33554432, 1073741824, 8589934592, 68719476736, 1099511627776, 17592186044416, 140737488355328, 1125899906842624, 72057594037927936, 4611686018427387904, 36893488147419103232, 295147905179352825856, 4722366482869645213696 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The numerators seems to coincide with A161736(n+2). 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). 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. The present partial sums are for H(1) with value 1/Gamma(3/2)^2 = 4/Pi (A088538). REFERENCES 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. LINKS FORMULA 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. MATHEMATICA Table[Denominator@ Sum[(1/(k + 1)) (Pochhammer[1/2, k]/k!)^2, {k, 0, n}], {n, 0, 19}] (* or *) Table[Denominator@ Sum[(1/(k + 1)) (Binomial[-1/2, k])^2, {k, 0, n}], {n, 0, 19}] (* or *) Table[Denominator@ Sum[(1/(k + 1)) ((2 k - 1)!!/(2 k)!!)^2, {k, 0, n}], {n, 0, 19}] (* Michael De Vlieger, Nov 15 2016 *) CROSSREFS Cf. A000165, A001147, A088538, A161736. Sequence in context: A087138 A293144 A111984 * A287230 A154710 A069033 Adjacent sequences:  A278142 A278143 A278144 * A278146 A278147 A278148 KEYWORD nonn,frac,easy AUTHOR Wolfdieter Lang, Nov 14 2016 STATUS approved

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Last modified April 18 17:25 EDT 2021. Contains 343089 sequences. (Running on oeis4.)