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 A176199 #8 Jun 19 2024 09:32:39
%S A176199 1,1,1,1,35,1,1,329,329,1,1,2535,6811,2535,1,1,18225,103925,103925,
%T A176199 18225,1,1,127435,1384685,2868895,1384685,127435,1,1,881977,17115873,
%U A176199 64568761,64568761,17115873,881977,1,1,6089807,202236439,1283008495,2302094507,1283008495,202236439,6089807,1
%N A176199 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 = 4.
%H A176199 G. C. Greubel, <a href="/A176199/b176199.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A176199 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 = 4.
%F A176199 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 = 4.
%F A176199 T(n, n-k) = T(n, k).
%e A176199 Triangle begins as:
%e A176199 1;
%e A176199 1, 1;
%e A176199 1, 35, 1;
%e A176199 1, 329, 329, 1;
%e A176199 1, 2535, 6811, 2535, 1;
%e A176199 1, 18225, 103925, 103925, 18225, 1;
%e A176199 1, 127435, 1384685, 2868895, 1384685, 127435, 1;
%e A176199 1, 881977, 17115873, 64568761, 64568761, 17115873, 881977, 1;
%t A176199 m:=13;
%t A176199 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 A176199 f[n_,k_,q_]:= Coefficient[Series[p[x,n,q], {x,0,m+2}], x, k];
%t A176199 T[n_,k_,q_]:= f[n,k,q] - f[n,0,q] + 1;
%t A176199 Table[T[n,k,4], {n,0,m}, {k,0,n}]//Flatten
%o A176199 (Magma)
%o A176199 m:=13;
%o A176199 R<x>:=PowerSeriesRing(Integers(), m+2);
%o A176199 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 A176199 f:= func< n,k,q | Coefficient(R!( p(x,n,q) ), k) >;
%o A176199 T:= func< n,k,q | f(n,k,q) - f(n,0,q) + 1 >; // T = A176199
%o A176199 [T(n,k,4): k in [0..n], n in [0..m]]; // _G. C. Greubel_, Jun 18 2024
%o A176199 (SageMath)
%o A176199 m=13
%o A176199 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 A176199 def f(n,k,q): return ( p(x,n,q) ).series(x, n+1).list()[k]
%o A176199 def T(n,k,q): return f(n,k,q) - f(n,0,q) + 1 # T = A176199
%o A176199 flatten([[T(n,k,4) for k in range(n+1)] for n in (0..m)]) # _G. C. Greubel_, Jun 18 2024
%Y A176199 Related triangles dependent on q: A008518 (q=1), A176198 (q=2), A174599 (q=3), this sequence (q=4).
%K A176199 nonn,tabl
%O A176199 0,5
%A A176199 _Roger L. Bagula_, Apr 11 2010
%E A176199 Edited by _G. C. Greubel_, Jun 18 2024