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 A206306 #24 Dec 21 2022 20:20:00
%S A206306 1,0,1,0,3,1,0,7,6,1,0,15,23,9,1,0,31,72,48,12,1,0,63,201,198,82,15,1,
%T A206306 0,127,522,699,420,125,18,1,0,255,1291,2223,1795,765,177,21,1,0,511,
%U A206306 3084,6562,6768,3840,1260,238,24,1
%N A206306 Riordan array (1, x/(1-3*x+2*x^2)).
%C A206306 The convolution triangle of the Mersenne numbers A000225. - _Peter Luschny_, Oct 09 2022
%H A206306 G. C. Greubel, <a href="/A206306/b206306.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A206306 Triangle T(n,k), read by rows, given by (0, 3, -2/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%F A206306 Diagonals sums are even-indexed Fibonacci numbers.
%F A206306 Sum_{k=0..n} T(n,k)*x^k = A000007(n), A204089(n), A204091(n) for x = 0, 1, 2 respectively.
%F A206306 G.f.: (1-3*x+2*x^)/(1-(3+y)*x+2*x^2).
%F A206306 From _Philippe Deléham_, Nov 17 2013; corrected Feb 13 2020: (Start)
%F A206306 T(n, n) = 1.
%F A206306 T(n+1, n) = 3n = A008585(n).
%F A206306 T(n+2, n) = A062725(n).
%F A206306 T(n,k) = 3*T(n-1,k)+T(n-1,k-1)-2*T(n-2,k), T(0,0)=T(1,1)=T(2,2)=1, T(1,0)=T(2,0)=0, T(2,1)=3, T(n,k)=0 if k<0 or if k>n. (End)
%F A206306 From _G. C. Greubel_, Dec 20 2022: (Start)
%F A206306 Sum_{k=0..n} (-1)^k*T(n,k) = [n=1] - A009545(n).
%F A206306 Sum_{k=0..n} (-2)^k*T(n,k) = [n=1] + A078020(n+1).
%F A206306 T(2*n, n+1) = A045741(n+2), n >= 0.
%F A206306 T(2*n+1, n+1) = A244038(n). (End)
%e A206306 Triangle begins:
%e A206306 1;
%e A206306 0, 1;
%e A206306 0, 3, 1;
%e A206306 0, 7, 6, 1;
%e A206306 0, 15, 23, 9, 1;
%e A206306 0, 31, 72, 48, 12, 1;
%e A206306 0, 63, 201, 198, 82, 15, 1;
%e A206306 0, 127, 522, 699, 420, 125, 18, 1;
%e A206306 0, 255, 1291, 2223, 1795, 765, 177, 21, 1;
%e A206306 0, 511, 3084, 6562, 6768, 3840, 1260, 238, 24, 1;
%e A206306 0, 1023, 7181, 18324, 23276, 16758, 7266, 1932, 308, 27, 1;
%p A206306 # Uses function PMatrix from A357368.
%p A206306 PMatrix(10, n -> 2^n - 1); # _Peter Luschny_, Oct 09 2022
%t A206306 T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==n, 1, If[k==0, 0, 3*T[n- 1, k] +T[n-1, k-1] -2*T[n-2, k]]]];
%t A206306 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 20 2022 *)
%o A206306 (Magma)
%o A206306 function T(n,k) // T = A206306
%o A206306 if k lt 0 or k gt n then return 0;
%o A206306 elif k eq n then return 1;
%o A206306 elif k eq 0 then return 0;
%o A206306 else return 3*T(n-1, k) +T(n-1, k-1) -2*T(n-2, k);
%o A206306 end if; return T;
%o A206306 end function;
%o A206306 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Dec 20 2022
%o A206306 (SageMath)
%o A206306 def T(n,k): # T = A206306
%o A206306 if (k<0 or k>n): return 0
%o A206306 elif (k==n): return 1
%o A206306 elif (k==0): return 0
%o A206306 else: return 3*T(n-1, k) +T(n-1, k-1) -2*T(n-2, k)
%o A206306 flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Dec 20 2022
%Y A206306 Columns: A000007, A000225 (Mersenne numbers), A045618, A055582.
%Y A206306 Cf. A008585, A009545, A084938, A062725, A110441, A204089, A204091.
%Y A206306 Cf. A045741, A078020, A244038.
%K A206306 easy,nonn,tabl
%O A206306 0,5
%A A206306 _Philippe Deléham_, Feb 06 2012