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 A165890 #10 Mar 10 2022 02:04:23
%S A165890 1,1,-2,1,1,0,-2,0,1,1,10,15,-52,15,10,1,1,44,484,-44,-970,-44,484,44,
%T A165890 1,1,150,5933,22792,466,-58684,466,22792,5933,150,1,1,472,58586,
%U A165890 682040,2085135,-682512,-4287444,-682512,2085135,682040,58586,472,1
%N A165890 Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = ( (1-x)^(n+1) * Sum_{k >= 0} (2*k+1)^(n-1)*x^k )^2, read by rows.
%H A165890 G. C. Greubel, <a href="/A165890/b165890.txt">Rows n = 0..50 of the irregular triangle, flattened</a>
%F A165890 T(n, k) = [x^k]( p(n, x) ), where p(n, x) = ( (1-x)^(n+1)*Sum_{k >= 0} (2*k+1)^(n-1)*x^k )^2.
%F A165890 T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (2^(n-1)*(1-x)^(n+2)*LerchPhi(x, -n+1, 1/2))^2.
%F A165890 Sum_{k=0..n} T(n, k) = 0^n.
%F A165890 T(n, n-k) = T(n, k). - _G. C. Greubel_, Mar 09 2022
%e A165890 Irregular triangle begins as:
%e A165890 1;
%e A165890 1, -2, 1;
%e A165890 1, 0, -2, 0, 1;
%e A165890 1, 10, 15, -52, 15, 10, 1;
%e A165890 1, 44, 484, -44, -970, -44, 484, 44, 1;
%e A165890 1, 150, 5933, 22792, 466, -58684, 466, 22792, 5933, 150, 1;
%t A165890 p[n_, x_]:= p[n, x]= If[n==0, 1, (2^(n-1)*(1-x)^(n+1)*LerchPhi[x, -n+1, 1/2])^2];
%t A165890 Table[CoefficientList[p[n, x], x], {n,0,12}]//Flatten (* modified by _G. C. Greubel_, Mar 09 2022 *)
%o A165890 (Sage)
%o A165890 def p(n,x): return (1-x)^(2*n+2)*sum( (2*j+1)^(n-1)*x^j for j in (0..2*n+2) )^2
%o A165890 def T(n,k): return ( p(n,x) ).series(x, 2*n+2).list()[k]
%o A165890 flatten([[T(n,k) for k in (0..2*n)] for n in (0..12)]) # _G. C. Greubel_, Mar 09 2022
%Y A165890 Cf. A000007 (row sums), A158782, A165883, A165889, A165891.
%K A165890 sign,tabf
%O A165890 0,3
%A A165890 _Roger L. Bagula_, Sep 29 2009
%E A165890 Edited by _G. C. Greubel_, Mar 09 2022