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A331322 a(n) = (3*n + 1)!/(n!)^3. 2
1, 24, 630, 16800, 450450, 12108096, 325909584, 8779605120, 236637794250, 6380456082000, 172080900531540, 4641917845743360, 125235075213284400, 3379123922914656000, 91184624634161304000, 2460769070127233057280, 66411927755894739034170, 1792432652235221330334000 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
FORMULA
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) = (-1)^n*A331431(2*n, n).
a(n) = (n+1)^2*A117671(n)*A000108(n). - G. C. Greubel, Mar 22 2022
From Karol A. Penson, Jul 28 2023: (Start)
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)
MAPLE
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);
# Karol A. Penson, Jul 28 2023
MATHEMATICA
Table[(3*n+1)*Binomial[3*n, n]*Binomial[2*n, n], {n, 0, 25}] (* G. C. Greubel, Mar 22 2022 *)
PROG
(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
CROSSREFS
Sequence in context: A358114 A097192 A182611 * A126153 A002553 A006201
KEYWORD
nonn
AUTHOR
Peter Luschny, Jan 18 2020
STATUS
approved

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Last modified March 29 09:32 EDT 2024. Contains 371268 sequences. (Running on oeis4.)