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A193554 Triangle read by rows: first column: top entry is 1, then powers of 2; rest of triangle is Pascal's triangle A007318.

%I A193554 #10 Sep 08 2022 08:45:58
%S A193554 1,1,1,2,1,1,4,1,2,1,8,1,3,3,1,16,1,4,6,4,1,32,1,5,10,10,5,1,64,1,6,
%T A193554 15,20,15,6,1,128,1,7,21,35,35,21,7,1,256,1,8,28,56,70,56,28,8,1,512,
%U A193554 1,9,36,84,126,126,84,36,9,1,1024,1,10,45,120,210,252,210,120,45,10,1,2048,1,11,55,165,330,462,462,330,165,55,11,1
%N A193554 Triangle read by rows: first column: top entry is 1, then powers of 2; rest of triangle is Pascal's triangle A007318.
%C A193554 The original definition of A135233 made no sense. In fact A135233 is the binomial transform of the present sequence.
%H A193554 G. C. Greubel, <a href="/A193554/b193554.txt">Rows n = 0..100 of triangle, flattened</a>
%e A193554 Triangle begins:
%e A193554    1;
%e A193554    1, 1;
%e A193554    2, 1, 1;
%e A193554    4, 1, 2,  1;
%e A193554    8, 1, 3,  3,  1;
%e A193554   16, 1, 4,  6,  4,  1;
%e A193554   32, 1, 5, 10, 10,  5, 1;
%e A193554   64, 1, 6, 15, 20, 15, 6, 1;
%e A193554 ...
%p A193554 T:= proc(n, k) option remember;
%p A193554       if k=n then 1
%p A193554     elif k=0 then 2^(n-1)
%p A193554     else binomial(n-1,k-1)
%p A193554       fi; end:
%p A193554 seq(seq(T(n, k), k=0..n), n=0..12); # _G. C. Greubel_, Nov 20 2019
%t A193554 T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==0, 2^(n-1), Binomial[n-1, k-1]]];
%t A193554 Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Nov 20 2019 *)
%o A193554 (PARI) T(n,k) = if(k==n, 1, if(k==0, 2^(n-1), binomial(n-1, k-1) )); \\ _G. C. Greubel_, Nov 20 2019
%o A193554 (Magma)
%o A193554 function T(n,k)
%o A193554   if k eq n then return 1;
%o A193554   elif k eq 0 then return 2^(n-1);
%o A193554   else return Binomial(n-1, k-1);
%o A193554   end if; return T; end function;
%o A193554 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 20 2019
%o A193554 (Sage)
%o A193554 @CachedFunction
%o A193554 def T(n, k):
%o A193554     if (k==n): return 1
%o A193554     elif (k==0): return 2^(n-1)
%o A193554     else: return binomial(n-1, k-1)
%o A193554 [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 20 2019
%Y A193554 Cf. A000079 (row sums)
%Y A193554 Cf. A007318, A135233.
%K A193554 nonn,tabl
%O A193554 0,4
%A A193554 _N. J. A. Sloane_, Jul 30 2011