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A174032
Triangle T(n, k) = Sum_{j>=0} floor(binomial(n, k)/2^j), read by rows.
3
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
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
KEYWORD
nonn,tabl,less,easy
AUTHOR
Roger L. Bagula, Mar 06 2010
STATUS
approved