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A278145
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Denominator of partial sums of the m=1 member of an m-family of series considered by Hardy with value 4/Pi (see A088538).
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3
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1, 8, 64, 1024, 16384, 131072, 1048576, 33554432, 1073741824, 8589934592, 68719476736, 1099511627776, 17592186044416, 140737488355328, 1125899906842624, 72057594037927936, 4611686018427387904, 36893488147419103232, 295147905179352825856, 4722366482869645213696
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OFFSET
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0,2
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COMMENTS
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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).
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REFERENCES
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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.
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LINKS
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FORMULA
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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STATUS
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approved
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