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 A113424 a(n) = (6*n)!/((3*n)!*(2*n)!*n!). 15
 1, 60, 13860, 4084080, 1338557220, 465817912560, 168470811709200, 62588625639883200, 23717177328413240100, 9124964373613212524400, 3553261127084984957001360, 1397224499394244497967972800, 553883078634868423069470550800, 221068174083308549543680044926400 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Appears in Ramanujan's theory of elliptic functions of signature 6. The family of elliptic curves "x=2*H=p^2+q^2-q^3, 0= 5 and positive integers n and k (apply Mestrovic, equation 39). (-1)^n*a(n) = [x^(2*n)*y^(2*n)] ( (1 + x + y)*(1 - x + y) )^(4*n). a(n) = [x^n] ( F(x) )^(60*n), where F(x) = 1 + x + 56*x^2 + 7355*x^3 + 1290319*x^4 + 264117464*x^5 + 59508459679*x^6 + ... appears to have integer coefficients. We conjecture that for k >= 1 the sequence defined by b_k(n) := [x^n] F(x)^(k*n) satisfies the above supercongruences for primes p >= 7. (End) From Peter Bala, Mar 20 2022: (Start) Right-hand side of the following identities valid for n >= 1: Sum_{k = 0..2*n} 4*n*(4*n+k-1)!/(k!*n!*(3*n)!) = (6*n)!/((3*n)!*(2*n)!*n!); Sum_{k = 0..3*n} 3*n*(3*n+k-1)!/(k!*n!*(2*n)!) = (6*n)!/((3*n)!(2*n)!*n!). Cf. A001451. (End) From Peter Bala, Feb 26 2023: (Start) a(n) = (4^n/n!^2) * Product_{k = n..3*n-1} 2*k + 1. a(n) = (12^n/n!^2) * Product_{k = 0..n-1} (6*k + 1)*(6*k + 5). (End) a(n) = 12*(6*n - 1)*(6*n - 5)*a(n-1)/n^2. - Neven Sajko, Jul 19 2023 From Karol A. Penson, Dec 26 2023: (Start) a(n) = Integral_{x=0..432} x^n*W(x) dx, n>=0, where W(x) = sqrt(18)*MeijerG([[], [0, 0]], [[-1/6, -5/6], []], x/432)/(1296*Pi), where MeijerG is the Meijer G - function. Apparently, W(x) cannot be represented by any other function. W(x) is positive on x = [0, 432], it diverges at x=0, and monotonically decreases for x>0. It appears that at x=432, W(x) tends to a constant value close to 0.000368414. This integral representation as the n-th power moment of the positive function W(x) on the interval [0, 432] is unique, as W(x) is the solution of the Hausdorff moment problem. (End) EXAMPLE G.f. = 1 + 60*x + 13860*x^2 + 4084080*x^3 + 1338557220*x^4 + ... - Michael Somos, Dec 02 2018 MATHEMATICA a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/6, 5/6, 1, 432 x], {x, 0, n}]; Table[Multinomial[n, 2 n, 3 n], {n, 0, 15}] (* Vladimir Reshetnikov, Oct 12 2016 *) a[ n_] := Multinomial[n, 2 n, 3 n]; (* Michael Somos, Dec 02 2018 *) PROG (PARI) {a(n) = if( n<0, 0, (6*n)! / ((3*n)! * (2*n)! * n!))}; (GAP) List([0..15], n->Factorial(6*n)/(Factorial(3*n)*Factorial(2*n)*Factorial(n))); # Muniru A Asiru, Apr 08 2018 CROSSREFS Elliptic Integrals: A002894, A006480, A000897. Factors: A005809, A066802. Sequence in context: A248708 A184890 A295598 * A009564 A269762 A291912 Adjacent sequences: A113421 A113422 A113423 * A113425 A113426 A113427 KEYWORD nonn,easy AUTHOR Michael Somos, Oct 31 2005 STATUS approved

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