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 A176204 #7 Sep 08 2022 08:45:52
%S A176204 1,1,1,1,13,1,1,41,41,1,1,101,261,101,1,1,225,1205,1205,225,1,1,477,
%T A176204 4761,9661,4761,477,1,1,985,17169,62473,62473,17169,985,1,1,2005,
%U A176204 58429,352933,624757,352933,58429,2005,1,1,4049,191357,1820765,5241413,5241413,1820765,191357,4049,1
%N A176204 Triangle T(n, k) = 4 * A008292(n+1, k) - 3, read by rows.
%C A176204 This sequence belongs to the class defined by T(n, m, q) = 2*T(n, m, q-1) - 1. The first few q values gives the sequences: A008292(n+1, k) (q=0), A176200 (q=1), this sequence (q=2).
%C A176204 Row sums are: {1, 2, 15, 84, 465, 2862, 20139, 161256, 1451493, 14515170, 159667167, ...}.
%C A176204 Former title: A recursive symmetrical triangular sequence based on Eulerian numbers: q=2: T(n, m, q) = 2*T(n, m, q-1) - 1.
%H A176204 G. C. Greubel, <a href="/A176204/b176204.txt">Rows n = 0..100 of the triangle, flattened</a>
%F A176204 T(n, m, q) = 2*T(n, m, q-1) - 1, with T(n, m, 0) = A008292(n+1, m).
%F A176204 From _G. C. Greubel_, Mar 12 2020: (Start)
%F A176204 T(n, k, q) = 2^q * A008292(n+1, k) - (2^q - 1).
%F A176204 Sum_{k=0..n} T(n, k, q) = (n+1)*( 2^q * n! - 2^q + 1) (row sums). (End)
%e A176204 Triangle begins as:
%e A176204 1;
%e A176204 1, 1;
%e A176204 1, 13, 1;
%e A176204 1, 41, 41, 1;
%e A176204 1, 101, 261, 101, 1;
%e A176204 1, 225, 1205, 1205, 225, 1;
%e A176204 1, 477, 4761, 9661, 4761, 477, 1;
%e A176204 1, 985, 17169, 62473, 62473, 17169, 985, 1;
%e A176204 1, 2005, 58429, 352933, 624757, 352933, 58429, 2005, 1;
%e A176204 1, 4049, 191357, 1820765, 5241413, 5241413, 1820765, 191357, 4049, 1;
%p A176204 A008292:= (n,k) -> add((-1)^j*binomial(n+1,j)*(k-j+1)^n, j=0..k+1);
%p A176204 A176204:= (n,k,q) -> 2^q*( A008292(n+1,k) -1) + 1;
%p A176204 seq(seq( A176204(n,k,2), k=0..n), n=0..12); # _G. C. Greubel_, Mar 12 2020
%t A176204 Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j,0,k+1}];
%t A176204 T[n_, m_, q_]:= 2^q*Eulerian[n+1, m] - 2^q +1;
%t A176204 Table[T[n, m, 2], {n,0,12}, {m,0,n}]//Flatten (* modified by _G. C. Greubel_, Mar 12 2020 *)
%o A176204 (PARI) Eulerian(n,k) = sum(j=0,k+1, (-1)^j*binomial(n+1,j)*(k-j+1)^n);
%o A176204 T(n,k,q) = 2^q*Eulerian(n+1,k) - (2^q - 1); \\ _G. C. Greubel_, Mar 12 2020
%o A176204 (Magma) Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]) >;
%o A176204 [[4*Eulerian(n+1,k) -3: k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, Mar 12 2020
%o A176204 (Sage)
%o A176204 def Eulerian(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j+1)^n for j in (0..k+1))
%o A176204 def T(n,k,q): return 2^q*Eulerian(n+1,k) - 2^q + 1
%o A176204 [[T(n,k,2) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Mar 12 2020
%Y A176204 Cf. A007318, A109128, A131061, A168625, A176200.
%K A176204 nonn,tabl
%O A176204 0,5
%A A176204 _Roger L. Bagula_, Apr 11 2010
%E A176204 Edited by _G. C. Greubel_, Mar 12 2020