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A203424
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Reciprocal of Vandermonde determinant of (1/2,1/4,...,1/(2n)).
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6
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1, -4, -144, 73728, 737280000, -183458856960000, -1381360067999170560000, 370806019753548356895375360000, 4086267719027580129096614223921807360000, -2092169072142121026097466482647965368320000000000000
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OFFSET
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1,2
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COMMENTS
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Each term divides its successor, as in A203425.
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LINKS
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FORMULA
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a(n) ~ (-1)^(n*(n-1)/2) * A * 2^(n^2/2 - n/2 - 1/2) * n^(n^2/2 - n/2 - 5/12) / (sqrt(Pi) * exp(n^2/4-n)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Dec 05 2015
a(n) = (-2)^binomial(n,2) * (n!)^n / BarnesG(n+2). - G. C. Greubel, Dec 07 2023
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MATHEMATICA
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(* First program *)
f[j_] := 1/(2 j); z = 16;
v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}];
1/Table[v[n], {n, z}] (* A203424 *)
Table[v[n]/(4 v[n + 1]), {n, z}] (* A203425 *)
(* Second program *)
Table[(-2)^Binomial[n, 2]*(n!)^n/BarnesG[n+2], {n, 20}] (* G. C. Greubel, Dec 07 2023 *)
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PROG
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(PARI) a(n) = prod(k=2, n, (-k)^(k-1)) << binomial(n, 2); \\ Kevin Ryde, May 03 2022
(Magma)
BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >;
A203424:= func< n| (-2)^Binomial(n, 2)*(Factorial(n))^n/BarnesG(n+2) >;
(SageMath)
def BarnesG(n): return product(factorial(k) for k in range(n-1))
def A203424(n): return (-2)^binomial(n, 2)*(gamma(n+1))^n/BarnesG(n+2)
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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