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 A168517 #6 Apr 01 2022 09:14:42
%S A168517 1,1,1,1,7,1,1,27,27,1,1,87,260,87,1,1,263,1828,1828,263,1,1,779,
%T A168517 11131,24746,11131,779,1,1,2299,62793,267515,267515,62793,2299,1,1,
%U A168517 6799,338902,2529377,4753074,2529377,338902,6799,1,1,20175,1780242,21935526,70068408,70068408,21935526,1780242,20175,1
%N A168517 Expansion of g.f. (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1-x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = -1, b = 1, and c = 1, read by rows.
%H A168517 G. C. Greubel, <a href="/A168517/b168517.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A168517 G.f.: (1/2)*( a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi(x, -n-1, 1) + c*2^(n+1)*(1 - x)^(n+1)*LerchPhi(x, -n, 1/2) ), where a = -1, b = 1, and c = 1.
%F A168517 From _G. C. Greubel_, Mar 31 2022: (Start)
%F A168517 T(n, k) = (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) ), with a = -1, b = 1, and c = 1.
%F A168517 T(n, n-k) = T(n, k). (End)
%e A168517 Triangle begins as:
%e A168517 1;
%e A168517 1, 1;
%e A168517 1, 7, 1;
%e A168517 1, 27, 27, 1;
%e A168517 1, 87, 260, 87, 1;
%e A168517 1, 263, 1828, 1828, 263, 1;
%e A168517 1, 779, 11131, 24746, 11131, 779, 1;
%e A168517 1, 2299, 62793, 267515, 267515, 62793, 2299, 1;
%e A168517 1, 6799, 338902, 2529377, 4753074, 2529377, 338902, 6799, 1;
%e A168517 1, 20175, 1780242, 21935526, 70068408, 70068408, 21935526, 1780242, 20175, 1;
%t A168517 p[x_, n_, a_, b_, c_]= (1/2)*(a*(1+x)^n + b*(1-x)^(n+2)*LerchPhi[x,-n-1,1] + c*2^(n+1)*(1-x)^(n+1)*LerchPhi[x,-n,1/2]);
%t A168517 Table[CoefficientList[p[x,n,-1,1,1], x], {n,0,10}]//Flatten (* modified by _G. C. Greubel_, Mar 31 2022 *)
%o A168517 (Sage)
%o A168517 def A168517(n,k,a,b,c): return (1/2)*( a*binomial(n,k) + sum( (-1)^(k-j)*(b*binomial(n+2, k-j)*(j+1)^(n+1) + 2*c*binomial(n+1,k-j)*(2*j+1)^n) for j in (0..k)) )
%o A168517 flatten([[A168517(n,k,-1,1,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 31 2022
%Y A168517 Cf. A168518, A168549, A168551, A168552.
%K A168517 nonn,tabl
%O A168517 0,5
%A A168517 _Roger L. Bagula_, Nov 28 2009
%E A168517 Edited by _G. C. Greubel_, Mar 31 2022