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 A324597 a(n) = n!^(4*n) * Product_{k=1..n} binomial(n + 1/k^3, n). 2
 1, 2, 918, 11592504000, 86712397842439769400000, 3472997049383321958747830928094241894400000, 4152034082374349458781848863476555783741415883758270213129361920000000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS In general, for m > 1, Product_{k=1..n} binomial(n + 1/k^m, n) ~ n^Zeta(m) / c(m), where c(m) = Product_{j>=1} Gamma(1 + 1/j^m)). Equivalently, c(m) = -gamma * Zeta(m) + Sum_{k>=2} (-1)^k*Zeta(k)*Zeta(m*k)/k, where gamma is the Euler-Mascheroni constant A001620. LINKS Table of n, a(n) for n=0..6. FORMULA a(n) ~ n!^(4*n) * n^Zeta(3) / (Product_{j>=1} Gamma(1 + 1/j^3)). a(n) ~ n^(4*n^2 + 2*n + Zeta(3)) * (2*Pi)^(2*n) / exp(4*n^2 - 1/3 - gamma*Zeta(3) + c), where c = A306778 = Sum_{k>=2} (-1)^k*Zeta(k)*Zeta(3*k)/k. MAPLE a:= n-> n!^(4*n)*mul(binomial(n+1/k^3, n), k=1..n): seq(a(n), n=0..7); # Alois P. Heinz, Jun 24 2023 MATHEMATICA Table[n!^(4*n) * Product[Binomial[n + 1/j^3, n], {j, 1, n}], {n, 1, 8}] CROSSREFS Cf. A306760, A324596, A306778. Sequence in context: A265881 A203609 A265617 * A159723 A282346 A070967 Adjacent sequences: A324594 A324595 A324596 * A324598 A324599 A324600 KEYWORD nonn AUTHOR Vaclav Kotesovec, Mar 09 2019 EXTENSIONS a(0)=1 prepended by Alois P. Heinz, Jun 24 2023 STATUS approved

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Last modified August 12 23:28 EDT 2024. Contains 375113 sequences. (Running on oeis4.)