A174037 Triangle T(n, k, q) = Sum_{j>=0} q^j * floor(binomial(n, k)/2^j) with q = 2, read by rows.

1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 12, 16, 12, 1, 1, 13, 36, 36, 13, 1, 1, 16, 49, 92, 49, 16, 1, 1, 17, 93, 197, 197, 93, 17, 1, 1, 32, 124, 304, 464, 304, 124, 32, 1, 1, 33, 204, 540, 768, 768, 540, 204, 33, 1, 1, 36, 237, 752, 1556, 1788, 1556, 752, 237, 36, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

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

Example

Triangle begins as:
  1;
  1,  1;
  1,  4,   1;
  1,  5,   5,   1;
  1, 12,  16,  12,    1;
  1, 13,  36,  36,   13,    1;
  1, 16,  49,  92,   49,   16,    1;
  1, 17,  93, 197,  197,   93,   17,   1;
  1, 32, 124, 304,  464,  304,  124,  32,   1;
  1, 33, 204, 540,  768,  768,  540, 204,  33,  1;
  1, 36, 237, 752, 1556, 1788, 1556, 752, 237, 36, 1;

Mathematica

T[n_, k_, q_]:= Sum[q^j*Floor[Binomial[n, k]/2^j], {j, 0, 2*n}];
Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 16 2021 *)

Prog

(Magma)
T:= func< n, k, q | (&+[q^j*Floor(Binomial(n, k)/2^j): j in [0..2*n]]) >;
[T(n, k, 2): 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, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 16 2021

Crossrefs

Cf. A174032 (q=1), this sequence (q=2), A174038 (q=3).

Sequence in context: A136489 A166455 A171142 * A173077 A131239 A114033

Adjacent sequences: A174034 A174035 A174036 * A174038 A174039 A174040

Keyword

nonn,tabl,less,easy

Author

Roger L. Bagula, Mar 06 2010

Extensions

Edited by G. C. Greubel, Apr 16 2021

Status

approved