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 A165883 #9 Mar 09 2022 01:39:08
%S A165883 1,1,2,1,1,10,26,10,1,1,34,287,508,287,34,1,1,102,2272,11098,19134,
%T A165883 11098,2272,102,1,1,294,15493,169432,675706,1042948,675706,169432,
%U A165883 15493,294,1,1,842,98374,2151026,17138559,55643460,82178676,55643460,17138559,2151026,98374,842,1
%N A165883 Irregular triangle T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1-x)^(2*n+3)*( Sum_{j >= 0} (2*j+ 1)^n*x^j )*( Sum_{j >= 0} j^(n+1)*x^j ), read by rows.
%H A165883 G. C. Greubel, <a href="/A165883/b165883.txt">Rows n = 0..50 of the irregular triangle, flattened</a>
%F A165883 T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (1-x)^(2*n+3)*( Sum_{j >= 0} (2*j+ 1)^n*x^j )*( Sum_{j >= 0} j^(n+1)*x^j ).
%F A165883 T(n, k) = [x^k]( p(n, x) ), where p(n, x) = 2^n*(1-x)^(2*n+3)*LerchPhi(x, -n, 1/2)*PolyLog(-n-1, x)/x.
%F A165883 T(n, n-k) = T(n, k). - _G. C. Greubel_, Mar 08 2022
%e A165883 Irregular triangle begins as:
%e A165883 1;
%e A165883 1, 2, 1;
%e A165883 1, 10, 26, 10, 1;
%e A165883 1, 34, 287, 508, 287, 34, 1;
%e A165883 1, 102, 2272, 11098, 19134, 11098, 2272, 102, 1;
%e A165883 1, 294, 15493, 169432, 675706, 1042948, 675706, 169432, 15493, 294, 1;
%t A165883 p[n_, x_]:= p[n, x]= (1/x)*(1-x)^(2*n+3)*Sum[(2*k+1)^n*x^k, {k,0,Infinity}]*Sum[k^(n+1)*x^k, {k,0,Infinity}];
%t A165883 Table[CoefficientList[p[n, x], x], {n,0,10}]//Flatten (* modified by _G. C. Greubel_, Mar 08 2022 *)
%o A165883 (Sage)
%o A165883 def p(n,x): return (1/x)*(1-x)^(2*n+3)*sum( (2*j+1)^n*x^j for j in (0..2*n+3) )*sum( j^(n+1)*x^j for j in (0..2*n+3) )
%o A165883 def T(n,k): return ( p(n,x) ).series(x, 2*n+1).list()[k]
%o A165883 flatten([[T(n,k) for k in (0..2*n)] for n in (0..12)]) # _G. C. Greubel_, Mar 08 2022
%Y A165883 Cf. A158782, A165889, A165890, A165891.
%K A165883 nonn,tabf
%O A165883 0,3
%A A165883 _Roger L. Bagula_, Sep 29 2009
%E A165883 Edited by _G. C. Greubel_, Mar 08 2022