A174032 Triangle T(n, k) = Sum_{j>=0} floor(binomial(n, k)/2^j), read by rows.

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 7, 10, 7, 1, 1, 8, 18, 18, 8, 1, 1, 10, 26, 38, 26, 10, 1, 1, 11, 39, 67, 67, 39, 11, 1, 1, 15, 53, 109, 137, 109, 53, 15, 1, 1, 16, 70, 165, 246, 246, 165, 70, 16, 1, 1, 18, 86, 236, 416, 498, 416, 236, 86, 18, 1

Offset

0,5

Links

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

Formula

T(n, k) = Sum_{j>=0} floor(binomial(n, k)/2^j).

T(n, k, q) = Sum_{j=0..2*n} q^j * floor(binomial(n,k)/2^j) for q = 1. - G. C. Greubel, Apr 16 2021

Example

Triangle begins as:
  1;
  1,  1;
  1,  3,  1;
  1,  4,  4,   1;
  1,  7, 10,   7,   1;
  1,  8, 18,  18,   8,   1;
  1, 10, 26,  38,  26,  10,   1;
  1, 11, 39,  67,  67,  39,  11,   1;
  1, 15, 53, 109, 137, 109,  53,  15,  1;
  1, 16, 70, 165, 246, 246, 165,  70, 16,  1;
  1, 18, 86, 236, 416, 498, 416, 236, 86, 18, 1;

Mathematica

T[n_, k_]:= Sum[Floor[Binomial[n, k]/2^j], {j, 0, 2*n}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Nov 28 2019 *)

Prog

(PARI) T(n, k) = round(suminf(j=0, binomial(n, k)\2^j )); \\ G. C. Greubel, Nov 28 2019
(Magma)
T:= func< n, k, q | (&+[q^j*Floor(Binomial(n, k)/2^j): j in [0..2*n]]) >;
[T(n, k, 1): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 16 2021
(Sage)
def T(n, k, q): return sum(q^j*( binomial(n, k)//2^j ) for j in (0..2*n))
flatten([[T(n, k, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 16 2021

Crossrefs

Cf. this sequence (q=1), A174037 (q=2), A174038 (q=3).

Sequence in context: A026648 A026747 A026374 * A180979 A102716 A173076

Adjacent sequences: A174029 A174030 A174031 * A174033 A174034 A174035

Keyword

nonn,tabl,less,easy

Author

Roger L. Bagula, Mar 06 2010

Status

approved