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

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, 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, 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

Offset: 0

Author

N. J. A. Sloane, Jul 30 2011

Keywords

• nonn
• tabl

Comments

The original definition of A135233 made no sense. In fact A135233 is the binomial transform of the present sequence.

Examples

			Triangle begins:
   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, 15, 20, 15, 6, 1;
...
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A000079 (row sums)

Cf. A007318, A135233.

Programs

Magma

function T(n,k)
  if k eq n then return 1;
  elif k eq 0 then return 2^(n-1);
  else return Binomial(n-1, k-1);
  end if; return T; end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019

Maple

T:= proc(n, k) option remember;
      if k=n then 1
    elif k=0 then 2^(n-1)
    else binomial(n-1,k-1)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019

Mathematica

T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==0, 2^(n-1), Binomial[n-1, k-1]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

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

Sage

@CachedFunction
def T(n, k):
    if (k==n): return 1
    elif (k==0): return 2^(n-1)
    else: return binomial(n-1, k-1)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019