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

0,4

Comments

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

Links

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

Example

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;
...

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 *)

Prog

(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
(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
(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

Crossrefs

Cf. A000079 (row sums)

Cf. A007318, A135233.

Sequence in context: A199856 A301906 A302150 * A131350 A131087 A105475

Adjacent sequences: A193551 A193552 A193553 * A193555 A193556 A193557

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jul 30 2011

Status

approved