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A331322
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a(n) = (3*n + 1)!/(n!)^3.
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2
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1, 24, 630, 16800, 450450, 12108096, 325909584, 8779605120, 236637794250, 6380456082000, 172080900531540, 4641917845743360, 125235075213284400, 3379123922914656000, 91184624634161304000, 2460769070127233057280, 66411927755894739034170, 1792432652235221330334000
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = [x^n] hypergeom([2/3, 4/3], [1], 27*x).
a(n) = 3*(9 - n^(-2))*a(n-1) for n > 0.
a(n) = Integral_{x=0..27} x^n*W(x) dx, where the weight function W(x) is defined on (0, 27) and it can be expressed with the Meijer G-function MeijerG as: W(x) = (sqrt(3)/(18*Pi))*MeijerG([[],[0,0]],[[-1/3,1/3],[]],x/27). The function W(x) is positive on its support (0, 27), is singular at x=0, and decreases monotonically to zero at x = 27.
The function W(x) is unique as it is the solution of the Hausdorff moment problem with the moments a(n). Due to the presence of two equal parameters (0,0) in MeijerG, it is not certain if W(x) can be represented by other known special functions. (End)
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MAPLE
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a := n -> (3*n+1)!/(n!)^3: seq(a(n), n=0..17); # Or:
hypergeom([2/3, 4/3], [1], 27*x): ser := series(%, x, 20):
seq(coeff(%, x, n), n=0..17); # Or:
a := proc(n) option remember; if n=0 then 1 else 3*(9 - n^(-2))*a(n-1) fi end:
# 4th Maple program:
W:=proc(x)sqrt(3)*MeijerG([[], [0, 0]], [[1/3, -1/3], []], x/27)/(18*Pi); end;
a:=proc(n) round(evalf[32](int(x^n*W(x), x=0..27))); end;
seq(a(n), n=0..17);
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MATHEMATICA
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Table[(3*n+1)*Binomial[3*n, n]*Binomial[2*n, n], {n, 0, 25}] (* G. C. Greubel, Mar 22 2022 *)
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PROG
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(Magma) [(n+1)^2*Binomial(3*n+1, n+1)*Catalan(n): n in [0..25]]; // G. C. Greubel, Mar 22 2022
(Sage) [(3*n+1)*binomial(2*n, n)*binomial(3*n, n) for n in (0..25)] # G. C. Greubel, Mar 22 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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