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 A174599 #14 Jun 19 2024 08:10:58
%S A174599 1,1,1,1,22,1,1,145,145,1,1,780,2246,780,1,1,3919,25144,25144,3919,1,
%T A174599 1,19202,243047,524812,243047,19202,1,1,93349,2168107,8760511,8760511,
%U A174599 2168107,93349,1,1,453592,18445564,127880680,235517062,127880680,18445564,453592,1
%N A174599 Triangle T(n, k) = A154646(n,k) - A154646(n,0) + 1, 0 <= k <= n.
%C A174599 The first and last element of each row of A154646 are reduced to 1 by subtracting a constant from each row.
%H A174599 G. C. Greubel, <a href="/A174599/b174599.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A174599 From _G. C. Greubel_, Jun 18 2024: (Start)
%F A174599 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 = 3.
%F A174599 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 = 3.
%F A174599 T(n, k) = A154646(n,k) - A154646(n,0) + 1.
%F A174599 T(n, n-k) = T(n, k). (End)
%e A174599 Triangle begins as:
%e A174599 1;
%e A174599 1, 1;
%e A174599 1, 22, 1;
%e A174599 1, 145, 145, 1;
%e A174599 1, 780, 2246, 780, 1;
%e A174599 1, 3919, 25144, 25144, 3919, 1;
%e A174599 1, 19202, 243047, 524812, 243047, 19202, 1;
%e A174599 1, 93349, 2168107, 8760511, 8760511, 2168107, 93349, 1;
%t A174599 m:=13;
%t A174599 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 A174599 f[n_,k_,q_]:= Coefficient[Series[p[x,n,q], {x,0,m+2}], x, k];
%t A174599 T[n_,k_,q_]:= f[n,k,q] -f[n,0,q] +1;
%t A174599 Table[T[n,k,3], {n,0,m}, {k,0,n}]//Flatten
%o A174599 (Magma)
%o A174599 m:=13;
%o A174599 R<x>:=PowerSeriesRing(Integers(), m+2);
%o A174599 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 A174599 f:= func< n,k,q | Coefficient(R!( p(x,n,q) ), k) >;
%o A174599 T:= func< n,k,q | f(n,k,q) -f(n,0,q) +1 >; // T = A174599
%o A174599 [T(n,k,3): k in [0..n], n in [0..m]]; // _G. C. Greubel_, Jun 18 2024
%o A174599 (SageMath)
%o A174599 m=13
%o A174599 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 A174599 def f(n,k,q): return ( p(x,n,q) ).series(x, n+1).list()[k]
%o A174599 def T(n,k,q): return f(n,k,q) - f(n,0,q) + 1 # T = A174599
%o A174599 flatten([[T(n,k,3) for k in range(n+1)] for n in (0..m)]) # _G. C. Greubel_, Jun 18 2024
%Y A174599 Related triangles dependent on q: A008518 (q=1), A176198 (q=2), this sequence (q=3), A176199 (q=4).
%Y A174599 Cf. A154646.
%K A174599 nonn,tabl
%O A174599 0,5
%A A174599 _Roger L. Bagula_, Mar 23 2010