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 A141720 #36 Feb 16 2025 08:33:08
%S A141720 0,1,0,1,0,1,2,-2,0,1,8,-8,0,1,22,-6,-32,16,0,1,52,84,-272,136,0,1,
%T A141720 114,606,-1168,-96,816,-272,0,1,240,2832,-2176,-8832,11904,-3968,0,1,
%U A141720 494,11122,11072,-83360,71168,13312,-31744,7936,0,1,1004,39772,148592,-472760,-17152,831232,-707584,176896
%N A141720 Triangle of coefficients of the polynomials (1 - x)^n*A(n,x/(1 - x)), where A(n,x) are the Eulerian polynomials of A008292.
%C A141720 Row sums are one.
%C A141720 Row n gives the coefficients in the expansion of Sum_{j=1..n} A008292(n,j)*x^j*(1 - x)^(n - j).
%C A141720 The coefficients of the polynomials (1 + x)^n*A(n,x/(1 + x)) are listed in A019538.
%H A141720 G. C. Greubel, <a href="/A141720/b141720.txt">Rows n = 0..50 of the irregular triangle, flattened</a>
%H A141720 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>
%F A141720 Row n is generated by the polynomial (1 - 2*x)^(n+1)*Li(-n, x/(1-x))/(1 - x), where Li(n, z) is the polylogarithm function.
%F A141720 Also generated by Sum_{k=0..n} (eulerian(n,k)*Sum_{l=0..n} (-1)^l*(n - l + 1)*(2 - x)^l*C(l + 1, k)). - Mourad Rahmani (mrahmani(AT)usthb.dz), Jul 22 2010
%F A141720 E.g.f.: (x*exp(2*x*y) - x*exp(y))/(x*exp(y) - (1 - x)*exp(2*x*y)). - _Franck Maminirina Ramaharo_, Oct 24 2018
%e A141720 Triangle begins:
%e A141720 0, 1;
%e A141720 0, 1;
%e A141720 0, 1, 2, -2;
%e A141720 0, 1, 8, -8;
%e A141720 0, 1, 22, -6, -32, 16;
%e A141720 0, 1, 52, 84, -272, 136;
%e A141720 0, 1, 114, 606, -1168, -96, 816, -272;
%e A141720 0, 1, 240, 2832, -2176, -8832, 11904, -3968;
%e A141720 0, 1, 494, 11122, 11072, -83360, 71168, 13312, -31744, 7936;
%e A141720 0, 1, 1004, 39772, 148592, -472760, -17152, 831232, -707584, 176896;
%e A141720 ...
%p A141720 CL := p -> PolynomialTools:-CoefficientList(p,x): flatten := seq -> ListTools:-Flatten(seq): flatten([seq(CL(add(A008292(n,j)*x^j*(1-x)^(n-j), j=1..n)), n=1..10)]); # _Peter Luschny_, Oct 25 2018
%t A141720 Table[CoefficientList[FullSimplify[(1-2x)^(1+n)*PolyLog[-n, x/(1-x)]/(1-x)], x], {n, 1, 10}]//Flatten
%o A141720 (Sage)
%o A141720 def A(n, k): return sum((-1)^j*binomial(n+1, j)*(k-j)^n for j in (0..k))
%o A141720 def p(n,x): return sum( A(n, j)*x^j*(1-x)^(n-j) for j in (0..n) )
%o A141720 def A141720(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
%o A141720 flatten([A141720(n) for n in range(1,13)]) # _G. C. Greubel_, Jul 15 2021
%o A141720 (Magma)
%o A141720 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A141720 f:= func< n,x | n eq 0 select 1 else (&+[EulerianNumber(n,j-1)*x^j*(1-x)^(n-j): j in [1..n]]) >;
%o A141720 A141720:= func< n,k | Coefficient(R!( f(n,x) ), k) >;
%o A141720 [A141720(n,k): k in [0..2*Floor((n+1)/2)-1], n in [1..15]]; // _G. C. Greubel_, Dec 30 2024
%Y A141720 Cf. A008292, A019538, A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221, A144387, A144400, A174128.
%K A141720 sign,tabf
%O A141720 1,7
%A A141720 _Roger L. Bagula_, Sep 11 2008
%E A141720 Edited by _Peter Bala_, Jul 04 2012
%E A141720 Edited, and extra term removed by _Franck Maminirina Ramaharo_, Oct 24 2018