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A059415
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Numerators of sequence arising from Apery's proof that zeta(3) is irrational.
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8
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0, 6, 351, 62531, 11424695, 35441662103, 20637706271, 963652602684713, 43190915887542721, 1502663969043851254939, 43786938951280269198311, 13780864457900933987428453, 51520703555193710949642777493
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OFFSET
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0,2
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REFERENCES
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M. Kontsevich and D. Zagier, Periods, pp. 771-808 of B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001.
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LINKS
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FORMULA
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(n+1)^3*a(n+1) = (34*n^3 + 51*n^2 + 27*n +5)*a(n) - n^3*a(n-1), n >= 1.
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EXAMPLE
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0, 6, 351/4, 62531/36, ...
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MAPLE
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a := proc(n) option remember; if n=0 then 0 elif n=1 then 6 else (n^(-3))* ( (34*(n-1)^3 + 51*(n-1)^2 + 27*(n-1) +5)*a((n-1)) - (n-1)^3*a((n-1)-1)); fi; end;
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MATHEMATICA
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a[n_] := Sum[ Binomial[n, k]^2*Binomial[k + n, k]^2*(Sum[1/m^3, {m, 1, n}] + Sum[(-1)^(m - 1)/(2*m^3*Binomial[n, m]*Binomial[m + n, m]), {m, 1, k}]), {k, 0, n}]; Table[a[n] // Numerator, {n, 0, 12}] (* Jean-François Alcover, Jul 16 2013, from the non-recursive formula *)
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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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