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 A182042 #20 Feb 17 2020 21:20:01
%S A182042 1,0,3,0,6,9,0,9,27,27,0,12,54,108,81,0,15,90,270,405,243,0,18,135,
%T A182042 540,1215,1458,729,0,21,189,945,2835,5103,5103,2187,0,24,252,1512,
%U A182042 5670,13608,20412,17496,6561,0,27,324,2268,10206,30618,61236,78732,59049,19683
%N A182042 Triangle T(n,k), read by rows, given by (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, 0, -3/2, 3/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%C A182042 Row sums are 4^n - 1 + 0^n.
%C A182042 Triangle of coefficients in expansion of (1+3*x)^n - 1 + 0^n.
%H A182042 G. C. Greubel, <a href="/A182042/b182042.txt">Rows n = 0..100 of triangle, flattened</a>
%F A182042 T(n,0) = 0^n; T(n,k) = binomial(n,k)*3^k for k > 0.
%F A182042 G.f.: (1-2*x+x^2+3*y*x^2)/(1-2*x-3*y*x+x^2+3*y*x^2).
%F A182042 T(n,k) = 2*T(n-1,k) + 3*T(n-1,k-1) - T(n-2,k) -3*T(n-2,k-1), T(0,0) = 1, T(1,0) = T(2,0) = 0, T(1,1) = 3, T(2,1) = 6, T(2,2) = 9 and T(n,k) = 0 if k < 0 or if k > n.
%F A182042 T(n,k) = A206735(n,k)*3^k.
%F A182042 T(n,k) = A013610(n,k) - A073424(n,k).
%e A182042 Triangle begins:
%e A182042 1;
%e A182042 0, 3;
%e A182042 0, 6, 9;
%e A182042 0, 9, 27, 27;
%e A182042 0, 12, 54, 108, 81;
%e A182042 0, 15, 90, 270, 405, 243;
%e A182042 0, 18, 135, 540, 1215, 1458, 729;
%e A182042 0, 21, 189, 945, 2835, 5103, 5103, 2187;
%p A182042 T:= proc(n, k) option remember;
%p A182042 if k=n then 3^n
%p A182042 elif k=0 then 0
%p A182042 else binomial(n,k)*3^k
%p A182042 fi; end:
%p A182042 seq(seq(T(n, k), k=0..n), n=0..10); # _G. C. Greubel_, Feb 17 2020
%t A182042 With[{m = 9}, CoefficientList[CoefficientList[Series[(1-2*x+x^2+3*y*x^2)/(1-2*x-3*y*x+x^2+3*y*x^2), {x, 0 , m}, {y, 0, m}], x], y]] // Flatten (* _Georg Fischer_, Feb 17 2020 *)
%o A182042 (PARI) T(n,k) = if (k==0, 1, binomial(n,k)*3^k);
%o A182042 matrix(10, 10, n, k, T(n-1,k-1)) \\ to see the triangle \\ _Michel Marcus_, Feb 17 2020
%o A182042 (Sage)
%o A182042 @CachedFunction
%o A182042 def T(n, k):
%o A182042 if (k==n): return 3^n
%o A182042 elif (k==0): return 0
%o A182042 else: return binomial(n,k)*3^k
%o A182042 [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Feb 17 2020
%Y A182042 Cf. A000244, A007318, A013610, A193193, A193194.
%K A182042 easy,nonn,tabl
%O A182042 0,3
%A A182042 _Philippe Deléham_, Apr 07 2012
%E A182042 a(48) corrected by _Georg Fischer_, Feb 17 2020