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 A135233 #15 Mar 27 2022 19:03:46
%S A135233 1,2,1,5,3,1,14,7,5,1,41,15,17,7,1,122,31,49,31,9,1,365,63,129,111,49,
%T A135233 11,1,1094,127,321,351,209,71,13,1,3281,255,769,1023,769,351,97,15,1,
%U A135233 9842,511,1793,2815,2561,1471,545,127,17,1
%N A135233 Triangle A007318 * A193554, read by rows.
%C A135233 Row sums = 3^n.
%C A135233 Left column = A007051: (1, 2, 5, 14, 41, 122, ...).
%H A135233 G. C. Greubel, <a href="/A135233/b135233.txt">Rows n = 0..100 of triangle, flattened</a>
%F A135233 Binomial transform of A193554, as infinite lower triangular matrices.
%F A135233 T(n,k) = Sum_{j=0..n-k} (-1)^(n-k+j)*binomial(n,j)*2^j, with T(n,n) = 1, and T(n,0) = (3^n + 1)/2. - _G. C. Greubel_, Nov 20 2019
%e A135233 First few rows of the triangle:
%e A135233 1;
%e A135233 2, 1;
%e A135233 5, 3, 1;
%e A135233 14, 7, 5, 1;
%e A135233 41, 15, 17, 7, 1;
%e A135233 ...
%p A135233 T:= proc(n, k) option remember;
%p A135233 if k=n then 1
%p A135233 elif k=0 then (3^n_1)/2
%p A135233 else add((-1)^(n-k+j)*binomial(n, j)*2^j, j=0..n-k)
%p A135233 fi; end:
%p A135233 seq(seq(T(n, k), k=0..n), n=0..12); # _G. C. Greubel_, Nov 20 2019
%t A135233 T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==0, (3^n+1)/2, Sum [(-1)^(n-k+i)* Binomial[n, i]*2^i, {i, 0, n-k}]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Nov 20 2019 *)
%o A135233 (PARI) T(n,k) = if(k==n, 1, if(k==0, (3^n+1)/2, sum(j=0, n-k, (-1)^(n-k+j)*binomial(n,j)*2^j) )); \\ _G. C. Greubel_, Nov 20 2019
%o A135233 (Magma)
%o A135233 function T(n,k)
%o A135233 if k eq n then return 1;
%o A135233 elif k eq 0 then return (3^n+1)/2;
%o A135233 else return (&+[(-1)^(n-k+j)*2^j*Binomial(n, j): j in [0..n-k]]);
%o A135233 end if; return T; end function;
%o A135233 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 20 2019
%o A135233 (Sage)
%o A135233 @CachedFunction
%o A135233 def T(n, k):
%o A135233 if (k==n): return 1
%o A135233 elif (k==0): return (3^n+1)/2
%o A135233 else: return sum((-1)^(n-k+j)*2^j*binomial(n, j) for j in (0..n-k))
%o A135233 [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 20 2019
%Y A135233 Cf. A007051, A007318, A118801, A119258, A193554.
%K A135233 nonn,tabl
%O A135233 0,2
%A A135233 _Gary W. Adamson_, Nov 23 2007
%E A135233 Definition corrected by _N. J. A. Sloane_, Jul 30 2011