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 A147563 #18 Mar 04 2023 02:02:04
%S A147563 4,4,4,4,-2,4,16,-8,4,44,-6,-16,4,4,104,84,-136,34,4,228,606,-584,-24,
%T A147563 102,-17,4,480,2832,-1088,-2208,1488,-248,4,988,11122,5536,-20840,
%U A147563 8896,832,-992,124,4,2008,39772,74296,-118190,-2144,51952,-22112,2764
%N A147563 Irregular triangle, T(n, k) = [x^k] p(n, x), where p(n, x) = 4*Sum_{j=0..n} A008292(n+1, j) * (x/2)^j * (1-x/2)^(n-j), read by rows.
%H A147563 G. C. Greubel, <a href="/A147563/b147563.txt">Row n = 0..50 of the irregular triangle, flattened</a>
%F A147563 T(n, k) = coefficients [x^k]( p(n, x) ), where p(n, x) = 4*Sum_{j=0..n} A008292(n+1, j) * (x/2)^j * (1-x/2)^(n-j).
%F A147563 T(n, k) = round( (-1/2)^(k-2) * Sum_{j=0..k} (-1)^j*binomial(n-j, k-j) * A008292(n+1, j+1) ). - _G. C. Greubel_, Mar 03 2023
%e A147563 Irregular triangle begins as:
%e A147563 4;
%e A147563 4;
%e A147563 4, 4, -2;
%e A147563 4, 16, -8;
%e A147563 4, 44, -6, -16, 4;
%e A147563 4, 104, 84, -136, 34;
%e A147563 4, 228, 606, -584, -24, 102, -17;
%e A147563 4, 480, 2832, -1088, -2208, 1488, -248;
%e A147563 4, 988, 11122, 5536, -20840, 8896, 832, -992, 124;
%e A147563 4, 2008, 39772, 74296, -118190, -2144, 51952, -22112, 2764;
%t A147563 (* First program *)
%t A147563 nmax:= 15;
%t A147563 p[x_, n_]= (1-x)^(n+1)*PolyLog[-n, x]/x;
%t A147563 b= Table[CoefficientList[p[x, n], x], {n, nmax+1}];
%t A147563 F[n_]:= CoefficientList[4*Sum[b[[n+1]][[m+1]]*(x/2)^(n-m)*(1-x/2)^m, {m, 0, n}], x];
%t A147563 T[n_]:= If[IntegerQ[F[n]], F[n], Sign[F[n]]*Abs[Round[F[n] - 1/2]]];
%t A147563 Table[T[n], {n, 0, nmax}]//Flatten
%t A147563 (* Second program *)
%t A147563 A008292[n_, k_]:= Sum[(-1)^j*(k-j)^n*Binomial[n+1,j], {j,0,k}];
%t A147563 F[n_, k_]:= (-1/2)^(k-2)*Sum[(-1)^j*Binomial[n-j, k-j]*A008292[n+1, j+ 1], {j,0,k}];
%t A147563 T[n_, k_]:= If[IntegerQ[F[n,k]], F[n,k], Sign[F[n,k]]*Abs[Round[F[n, k] - 1/2]]];
%t A147563 Table[T[n, k], {n,0,16}, {k, 0, 2*Floor[n/2]}]//Flatten (* _G. C. Greubel_, Mar 03 2023 *)
%o A147563 (Magma)
%o A147563 A008292:= func< n,k | (&+[(-1)^j*Binomial(n+1, j)*(k-j)^n: j in [0..k]]) >;
%o A147563 T:= func< n,k | (-1/2)^(k-2)*(&+[(-1)^j*Binomial(n-j,k-j)*A008292(n+1,j+1): j in [0..k]]) >;
%o A147563 [Floor(T(n,k)): k in [0..2*Floor(n/2)], n in [0..16]]; // _G. C. Greubel_, Oct 27 2022; Mar 03 2023
%o A147563 (SageMath)
%o A147563 def A008292(n,k): return sum( (-1)^j*binomial(n+1, j)*(k-j)^n for j in range(k+1) )
%o A147563 def A147563(n,k): return floor((-1/2)^(k-2)*sum((-1)^j*binomial(n-j, k-j)*A008292(n+1,j+1) for j in range(k+1)))
%o A147563 flatten([[A147563(n,k) for k in range(2*floor(n/2) + 1)] for n in range(16)]) # _G. C. Greubel_, Oct 27 2022; Mar 03 2023
%Y A147563 Cf. A008292.
%K A147563 sign,tabf
%O A147563 0,1
%A A147563 _Roger L. Bagula_, Nov 07 2008
%E A147563 Edited by _G. C. Greubel_, Oct 27 2022