This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A162990 #20 Feb 16 2025 08:33:11
%S A162990 4,36,9,576,144,64,14400,3600,1600,900,518400,129600,57600,32400,
%T A162990 20736,25401600,6350400,2822400,1587600,1016064,705600,1625702400,
%U A162990 406425600,180633600,101606400,65028096,45158400,33177600,131681894400
%N A162990 Triangle of polynomial coefficients related to 3F2([1,n+1,n+1],[n+2,n+2],z).
%C A162990 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 A162990 We observe that 3F2([1,1,1],[2,2],z) = Li2(z)/z and that 3F2([1,0,0],[1,1],z) = 1.
%C A162990 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 A162990 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 A162990 Lewin, L., Polylogarithms and Associated Functions. New York, North-Holland, 1981.
%H A162990 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 A162990 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Dilogarithm.html">Dilogarithm</a>.
%F A162990 a(n,m) = ((n+1)!/m)^2 for n >= 1 and 1 <= m <= n.
%e A162990 The first few rows of the triangle are:
%e A162990 [4]
%e A162990 [36, 9]
%e A162990 [576, 144, 64]
%e A162990 [14400, 3600, 1600, 900]
%e A162990 The first few MN(z;n) polynomials are:
%e A162990 MN(z;n=1) = 4
%e A162990 MN(z;n=2) = 36 + 9*z
%e A162990 MN(z;n=3) = 576 + 144*z + 64*z^2
%e A162990 MN(z;n=4) = 14400 + 3600*z + 1600*z^2 + 900*z^3
%p A162990 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 A162990 Table[((n+1)!/m)^2, {n, 10}, {m, n}] (* _Paolo Xausa_, Mar 30 2024 *)
%Y A162990 A162995 is a scaled version of this triangle.
%Y A162990 A001819(n)*(n+1)^2 equals the row sums for n>=1.
%Y A162990 A162991 and A162992 equal the first and second right hand columns.
%Y A162990 A001048, A052747, A052759, A052778, A052794 are related to the square root of the first five right hand columns.
%Y A162990 A001044, A162993 and A162994 equal the first, second and third left hand columns.
%Y A162990 A000142, A001710, A002301, A133799, A129923, A001715 are related to the square root of the first six left hand columns.
%Y A162990 A027451(n+1) equals the denominators of M(z, n)/(n!)^2.
%Y A162990 A129202(n)/A129203(n) = (n+1)^2*Li2(z=1)/(Pi^2) = (n+1)^2/6.
%Y A162990 Cf. A002378 and A035287.
%K A162990 easy,nonn,tabl
%O A162990 1,1
%A A162990 _Johannes W. Meijer_, Jul 21 2009