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Triangle of polynomial coefficients related to 3F2([1,n+1,n+1],[n+2,n+2],z).
14

%I #19 Mar 30 2024 21:19:52

%S 4,36,9,576,144,64,14400,3600,1600,900,518400,129600,57600,32400,

%T 20736,25401600,6350400,2822400,1587600,1016064,705600,1625702400,

%U 406425600,180633600,101606400,65028096,45158400,33177600,131681894400

%N Triangle of polynomial coefficients related to 3F2([1,n+1,n+1],[n+2,n+2],z).

%C 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.

%C We observe that 3F2([1,1,1],[2,2],z) = Li2(z)/z and that 3F2([1,0,0],[1,1],z) = 1.

%C 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)).

%C 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.

%D Lewin, L., Polylogarithms and Associated Functions. New York, North-Holland, 1981.

%H Paolo Xausa, <a href="/A162990/b162990.txt">Table of n, a(n) for n = 1..5050</a> (rows 1..100 of the triangle, flattened).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Dilogarithm.html">Dilogarithm</a>.

%F a(n,m) = ((n+1)!/m)^2 for n >= 1 and 1 <= m <= n.

%e The first few rows of the triangle are:

%e [4]

%e [36, 9]

%e [576, 144, 64]

%e [14400, 3600, 1600, 900]

%e The first few MN(z;n) polynomials are:

%e MN(z;n=1) = 4

%e MN(z;n=2) = 36 + 9*z

%e MN(z;n=3) = 576 + 144*z + 64*z^2

%e MN(z;n=4) = 14400 + 3600*z + 1600*z^2 + 900*z^3

%p 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

%t Table[((n+1)!/m)^2, {n, 10}, {m, n}] (* _Paolo Xausa_, Mar 30 2024 *)

%Y A162995 is a scaled version of this triangle.

%Y A001819(n)*(n+1)^2 equals the row sums for n>=1.

%Y A162991 and A162992 equal the first and second right hand columns.

%Y A001048, A052747, A052759, A052778, A052794 are related to the square root of the first five right hand columns.

%Y A001044, A162993 and A162994 equal the first, second and third left hand columns.

%Y A000142, A001710, A002301, A133799, A129923, A001715 are related to the square root of the first six left hand columns.

%Y A027451(n+1) equals the denominators of M(z, n)/(n!)^2.

%Y A129202(n)/A129203(n) = (n+1)^2*Li2(z=1)/(Pi^2) = (n+1)^2/6.

%Y Cf. A002378 and A035287.

%K easy,nonn,tabl

%O 1,1

%A _Johannes W. Meijer_, Jul 21 2009