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A100516
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Numerator of Sum_{k=0..n} 1/binomial(n,k)^2.
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8
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1, 2, 9, 20, 155, 21, 7441, 3224, 5697, 3575, 28523, 27183, 70357417, 4661447, 386395, 8959408, 10028928779, 525966759, 1476346738309, 35051863075, 847581175, 709068173, 62385202783, 20340152122, 119483756745025, 4418168441921, 311960929172031
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OFFSET
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0,2
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REFERENCES
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H. W. Gould, Combinatorial Identities, Morgantown, 1972, p. 50, formula (5.2).
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LINKS
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FORMULA
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a(n) = numerator( 3*(n+1)^2/((n+2)*(2*n+3)*Catalan(n+1)) * Sum_{k=1..n+1} binomial(2*k, k)/k ). - G. C. Greubel, Jun 24 2022
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EXAMPLE
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1, 2, 9/4, 20/9, 155/72, 21/10, 7441/3600, 3224/1575, 5697/2800, 3575/1764, 28523/14112, 27183/13475, 70357417/34927200, 4661447/2316600, ... = A100516/A100517
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MATHEMATICA
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Table[3*(n+1)^2/((n+2)*(2*n+3)*CatalanNumber[n+1])*Sum[((k+ 1)/k)*CatalanNumber[k], {k, n+1}], {n, 0, 40}]//Numerator (* G. C. Greubel, Jun 24 2022 *)
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PROG
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(Magma) [Numerator( (&+[1/Binomial(n, k)^2: k in [0..n]]) ): n in [0..40]]; // G. C. Greubel, Jun 24 2022
(SageMath) [numerator(sum(1/binomial(n, k)^2 for k in (0..n))) for n in (0..40)] # G. C. Greubel, Jun 24 2022
(PARI) a(n) = numerator(sum(k=0, n, 1/binomial(n, k)^2)); \\ Michel Marcus, Jun 24 2022
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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