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 A201972 #20 Feb 18 2020 01:18:41
%S A201972 1,2,2,5,8,3,12,28,20,4,29,88,94,40,5,70,262,372,244,70,6,169,752,
%T A201972 1333,1184,539,112,7,408,2104,4472,5016,3144,1064,168,8,985,5776,
%U A201972 14316,19408,15526,7344,1932,240,9
%N A201972 Triangle T(n,k), read by rows, given by (2,1/2,-1/2,0,0,0,0,0,0,0,...) DELTA (2,-1/2,1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.
%C A201972 Diagonal sums: A201967(n), row sums: A000302(n) (powers of 4).
%H A201972 G. C. Greubel, <a href="/A201972/b201972.txt">Rows n = 0..100 of triangle, flattened</a>
%F A201972 G.f.: 1/(1-2*(y+1)*x+(y+1)*(y-1)*x^2).
%F A201972 Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000129(n+1), A000302(n), A138395(n), A057084(n) for x = -1, 0, 1, 2, 3, respectively.
%F A201972 Sum_{k=0..n} T(n,k)*x^(n-k) = A000027(n), A000302(n), A090018(n), A057091(n) for x = 0, 1, 2, 3, respectively.
%F A201972 T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-2) with T(0,0) = 1, T(n,k) = 0 if k < 0 or if n < k.
%e A201972 Triangle begins:
%e A201972 1;
%e A201972 2, 2;
%e A201972 5, 8, 3;
%e A201972 12, 28, 20, 4;
%e A201972 29, 88, 94, 40, 5;
%e A201972 70, 262, 372, 244, 70, 6;
%e A201972 169, 752, 1333, 1184, 539, 112, 7;
%p A201972 T:= proc(n, k) option remember;
%p A201972 if k=0 and n=0 then 1
%p A201972 elif k<0 or k>n then 0
%p A201972 else 2*T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-2)
%p A201972 fi; end:
%p A201972 seq(seq(T(n, k), k=0..n), n=0..10); # _G. C. Greubel_, Feb 17 2020
%t A201972 With[{m = 8}, CoefficientList[CoefficientList[Series[1/(1-2*(y+1)*x+(y+1)*(y-1)*x^2), {x, 0 , m}, {y, 0, m}], x], y]] // Flatten (* _Georg Fischer_, Feb 17 2020 *)
%o A201972 (PARI) T(n,k) = if(n<k,0,if(k<0,0,if((n==0)&&(k==0),1,2*T(n-1,k)+2*T(n-1,k-1)+T(n-2,k)-T(n-2,k-2))));
%o A201972 matrix(10, 10, n, k, T(n-1,k-1)) \\ to see the triangle \\ _Michel Marcus_, Feb 17 2020
%o A201972 (Sage)
%o A201972 @CachedFunction
%o A201972 def T(n, k):
%o A201972 if (k<0 or k>n): return 0
%o A201972 elif (k==0 and n==0): return 1
%o A201972 else: return 2*T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-2)
%o A201972 [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Feb 17 2020
%Y A201972 Cf. A000027, A000129, A006645, A007290.
%K A201972 nonn,tabl
%O A201972 0,2
%A A201972 _Philippe Deléham_, Dec 07 2011
%E A201972 a(40) corrected by _Georg Fischer_, Feb 17 2020