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 A203424 Reciprocal of Vandermonde determinant of (1/2,1/4,...,1/(2n)). 6
 1, -4, -144, 73728, 737280000, -183458856960000, -1381360067999170560000, 370806019753548356895375360000, 4086267719027580129096614223921807360000, -2092169072142121026097466482647965368320000000000000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Each term divides its successor, as in A203425. LINKS G. C. Greubel, Table of n, a(n) for n = 1..35 FORMULA a(n) = Product_{k=1..n} (-2k)^(k-1). - Andrei Asinowski, Nov 03 2015 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) * A203421(n). - Kevin Ryde, May 03 2022 a(n) = (-2)^binomial(n,2) * (n!)^n / BarnesG(n+2). - G. C. Greubel, Dec 07 2023 MATHEMATICA (* 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 *) PROG (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) >; [A203424(n): n in [1..20]]; // G. C. Greubel, Dec 07 2023 (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) [A203424(n) for n in range(1, 21)] # G. C. Greubel, Dec 07 2023 CROSSREFS Cf. A203425, A203421. Sequence in context: A069135 A186081 A138176 * A055209 A239350 A343697 Adjacent sequences: A203421 A203422 A203423 * A203425 A203426 A203427 KEYWORD sign,easy AUTHOR Clark Kimberling, Jan 02 2012 STATUS approved

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Last modified June 24 16:27 EDT 2024. Contains 373679 sequences. (Running on oeis4.)