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A074800
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a(n) = denominator( (4*n+1)*(Product_{i=1..n} (2*i-1)/Product_{i=1..n} (2*i))^5 ).
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4
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1, 32, 32768, 1048576, 34359738368, 1099511627776, 1125899906842624, 36028797018963968, 37778931862957161709568, 1208925819614629174706176, 1237940039285380274899124224, 39614081257132168796771975168
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OFFSET
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0,2
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COMMENTS
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For the partial sums of the series given in the formula section see A278140(n)/a(n). The value of the series is given in A277235. - Wolfdieter Lang, Nov 15 2016
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REFERENCES
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Bruce C. Berndt and Robert Rankin,"Ramanujan: letters and commentary", AMS-LMS, History of Mathematics, vol. 9, p. 57
G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 7, 105-106, 111.
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LINKS
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FORMULA
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a(n) = denominator(b(n)) with b(0) = 1 and b(n) = (4*n+1)*(Product_{i=1..n} (2*i-1) / Product_{i=1..n}(2*i))^5 = (4*n+1)*(A001147(n)/A000165(n))^5.
1 + Sum_{k>=1} (-1)^k*b(k) = 2/Gamma(3/4)^4=0.88694116857811540541...(see
a(n) = denominator( (4*n+1)*( binomial(2*n, n)/4^n )^5 ). - G. C. Greubel, Jul 09 2021
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MATHEMATICA
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Table[Denominator[(4n+1) (Product[(2i-1), {i, n}]/Product[2i, {i, n}])^5], {n, 0, 10}] (* Michael De Vlieger, Nov 15 2016 *)
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PROG
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(PARI) a(n)=denominator ((4*n+1)*(prod(i=1, n, 2*i-1)/prod(i=1, n, 2*i))^5)
(Magma) [Denominator((4*n+1)*((n+1)*Catalan(n)/4^n)^5): n in [0..30]]; // G. C. Greubel, Jul 09 2021
(Sage) [denominator((4*n+1)*(binomial(2*n, n)/4^n)^5) for n in (0..30)] # G. C. Greubel, Jul 09 2021
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CROSSREFS
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KEYWORD
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easy,frac,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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