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 A176198 #8 Jun 19 2024 08:10:36
%S A176198 1,1,1,1,11,1,1,45,45,1,1,151,459,151,1,1,473,3363,3363,473,1,1,1443,
%T A176198 21085,47095,21085,1443,1,1,4357,121313,519445,519445,121313,4357,1,1,
%U A176198 13103,663223,4970575,9350027,4970575,663223,13103,1,1,39345,3512679,43415943,138826587,138826587,43415943,3512679,39345,1
%N A176198 Triangle, read by rows, T(n, k) = f(n,k,q) - f(n,0,q) + 1, where f(n, k, q) = [x^k](p(x,n,q)), p(x, n, q) = (1-x)^(n+1)*Sum_{k >= 0} ( (q*k+1)^n + (q*(k+1)-1)^n )*x^k, and q = 2.
%H A176198 G. C. Greubel, <a href="/A176198/b176198.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A176198 T(n, k) = f(n, k, q) - f(n, 0, q) + 1, where f(n, k, q) = [x^k]( p(x, n, q) ), p(x, n, q) = (1-x)^(n+1) * Sum_{k >= 0} ( (q*k + 1)^n + (q*(k+1) - 1)^n )*x^k, and q = 2.
%F A176198 T(n, k) f(n, k, q) - f(n, 0, q) + 1, where f(n, k, q) = [x^k]( p(x, n, q) ), p(x, n, q) = q^n * (1-x)^(n+1) * ( LerchPhi(x, -n, 1/q) + LerchPhi(x, -n, (q-1)/q) ), and q = 2.
%F A176198 T(n, n-k) = T(n, k).
%e A176198 Triangle begins as:
%e A176198 1;
%e A176198 1, 1;
%e A176198 1, 11, 1;
%e A176198 1, 45, 45, 1;
%e A176198 1, 151, 459, 151, 1;
%e A176198 1, 473, 3363, 3363, 473, 1;
%e A176198 1, 1443, 21085, 47095, 21085, 1443, 1;
%e A176198 1, 4357, 121313, 519445, 519445, 121313, 4357, 1;
%e A176198 1, 13103, 663223, 4970575, 9350027, 4970575, 663223, 13103, 1;
%t A176198 m:=13;
%t A176198 p[x_,n_,q_]:= (1-x)^(n+1)*Sum[((q*j+1)^n+(q*(j+1)-1)^n)*x^j, {j,0,m+ 2}];
%t A176198 f[n_,k_,q_]:= Coefficient[Series[p[x,n,q], {x,0,m+2}], x, k];
%t A176198 T[n_,k_,q_]:= f[n,k,q] - f[n,0,q] + 1; (* T = A176198 *)
%t A176198 Table[T[n,k,2], {n,0,m}, {k,0,n}]//Flatten
%o A176198 (Magma)
%o A176198 m:=13;
%o A176198 R<x>:=PowerSeriesRing(Integers(), m+2);
%o A176198 p:= func< x,n,q | (1-x)^(n+1)*(&+[((q*j+1)^n + (q*(j+1)-1)^n)*x^j: j in [0..m+2]]) >;
%o A176198 f:= func< n,k,q | Coefficient(R!( p(x,n,q) ), k) >;
%o A176198 T:= func< n,k,q | f(n,k,q) - f(n,0,q) + 1 >; // T = A176198
%o A176198 [T(n,k,2): k in [0..n], n in [0..m]]; // _G. C. Greubel_, Jun 18 2024
%o A176198 (SageMath)
%o A176198 m=13
%o A176198 def p(x,n,q): return (1-x)^(n+1)*sum(((q*j+1)^n + (q*(j+1)-1)^n)*x^j for j in range(m+3))
%o A176198 def f(n,k,q): return ( p(x,n,q) ).series(x, n+1).list()[k]
%o A176198 def T(n,k,q): return f(n,k,q) - f(n,0,q) + 1 # T = A176198
%o A176198 flatten([[T(n,k,2) for k in range(n+1)] for n in (0..m)]) # _G. C. Greubel_, Jun 18 2024
%Y A176198 Related triangles dependent on q: A008518 (q=1), this sequence (q=2), A174599 (q=3), A176199 (q=4).
%K A176198 nonn,tabl
%O A176198 0,5
%A A176198 _Roger L. Bagula_, Apr 11 2010
%E A176198 Edited by _G. C. Greubel_, Jun 19 2024