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 A108400 a(n) = Product_{k = 0..n} (2^k * k!). 9
 1, 2, 16, 768, 294912, 1132462080, 52183852646400, 33664847019245568000, 347485857744891213250560000, 64560982045934655213753964953600000, 239901585047846581083822477336190648320000000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Hankel transform (see A001906 for definition) of the sequences A000898, A001861, A035009(with first term omitted), A047974, A067147(unsigned version), A083886. Hankel transform of the sequence with e.g.f. exp(x^2). Also (-1)^C(n+1,2)*a(n) is the Hankel transform of the sequence with e.g.f. exp(-x^2). - Paul Barry, Feb 12 2008 Let T(n,k) = (n+1)^k * (1+(-1)^(n-k))/2, then a(n) = det(T(i,j); 0<=i, j<=n). - Paul Barry, Feb 12 2008 LINKS G. C. Greubel, Table of n, a(n) for n = 0..38 M. E. Larsen, Wronskian Harmony, Mathematics Magazine, vol. 63, no. 1, 1990, pp. 33-37. J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5. FORMULA a(n) = A006125(n+1)*A000178(n). a(n) = Product_{i=1..n} Product_{j=0..i-1} {2*(i-j)}. - Paul Barry, Aug 02 2008 a(n) ~ 2^((n+1)^2/2) * n^(n^2/2+n+5/12) * Pi^((n+1)/2) / (A * exp(3*n^2/4+n-1/12)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 14 2014 MATHEMATICA Table[Product[k!*2^k, {k, 0, n}], {n, 0, 10}] (* Vaclav Kotesovec, Nov 14 2014 *) Table[2^Binomial[n+1, 2]*BarnesG[n+2], {n, 0, 15}] (* G. C. Greubel, Jun 21 2022 *) PROG (Magma) BarnesG:= func< n | (&*[Factorial(j): j in [0..n-2]]) >; [2^Binomial(n+1, 2)*BarnesG(n+2): n in [0..15]]; // G. C. Greubel, Jun 21 2022 (SageMath) def barnes_g(n): return product(factorial(j) for j in (0..n-2)) [2^binomial(n+1, 2)*barnes_g(n+2) for n in (0..15)] # G. C. Greubel, Jun 21 2022 CROSSREFS Cf. A000178, A006125, A074962. Cf. A000898, A001861, A001906, A035009, A047974, A067147, A083886. Sequence in context: A278087 A015188 A005118 * A186002 A013029 A012915 Adjacent sequences: A108397 A108398 A108399 * A108401 A108402 A108403 KEYWORD nonn,easy AUTHOR Philippe Deléham, Jul 02 2005 STATUS approved

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Last modified September 26 12:34 EDT 2023. Contains 365657 sequences. (Running on oeis4.)