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A162990
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Triangle of polynomial coefficients related to 3F2([1,n+1,n+1],[n+2,n+2],z).
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14
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4, 36, 9, 576, 144, 64, 14400, 3600, 1600, 900, 518400, 129600, 57600, 32400, 20736, 25401600, 6350400, 2822400, 1587600, 1016064, 705600, 1625702400, 406425600, 180633600, 101606400, 65028096, 45158400, 33177600, 131681894400
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OFFSET
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1,1
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COMMENTS
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The hypergeometric function 3F2([1,n+1,n+1],[n+2,n+2],z) = (n+1)^2*Li2(z)/z^(n+1) - MN(z;n)/(n!^2*z^n) for n >= 1, with Li2(z) the dilogarithm. The polynomial coefficients of MN(z;n) lead to the triangle given above.
We observe that 3F2([1,1,1],[2,2],z) = Li2(z)/z and that 3F2([1,0,0],[1,1],z) = 1.
The generating function for the EG1[3,n] coefficients of the EG1 matrix, see A162005, is GFEG1(z;m=2) = 1/(1-z)*(3*zeta(3)/2-2*z*log(2)* 3F2([1,1,1],[2,2],z) + sum((2^(1-2*n)* factorial(2*n-1)*z^(n+1)*3F2([1,n+1,n+1],[n+2,n+2],z))/(factorial(n+1)^2), n=1..infinity)).
The zeros of the MN(z;n) polynomials for larger values of n get ever closer to the unit circle and resemble the full moon, hence we propose to call the MN(z;n) the moon polynomials.
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REFERENCES
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Lewin, L., Polylogarithms and Associated Functions. New York, North-Holland, 1981.
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LINKS
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FORMULA
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a(n,m) = ((n+1)!/m)^2 for n >= 1 and 1 <= m <= n.
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EXAMPLE
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The first few rows of the triangle are:
[4]
[36, 9]
[576, 144, 64]
[14400, 3600, 1600, 900]
The first few MN(z;n) polynomials are:
MN(z;n=1) = 4
MN(z;n=2) = 36 + 9*z
MN(z;n=3) = 576 + 144*z + 64*z^2
MN(z;n=4) = 14400 + 3600*z + 1600*z^2 + 900*z^3
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MAPLE
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a := proc(n, m): ((n+1)!/m)^2 end: seq(seq(a(n, m), m=1..n), n=1..7); # Johannes W. Meijer, revised Nov 29 2012
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MATHEMATICA
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Table[((n+1)!/m)^2, {n, 10}, {m, n}] (* Paolo Xausa, Mar 30 2024 *)
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CROSSREFS
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A162995 is a scaled version of this triangle.
A001819(n)*(n+1)^2 equals the row sums for n>=1.
A027451(n+1) equals the denominators of M(z, n)/(n!)^2.
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KEYWORD
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AUTHOR
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STATUS
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approved
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