A174097 Triangle T(n,k,q) = Sum_{j=0..10} q^j * floor(A174093(n,k)/2^j) with q=3, read by rows.

1, 1, 1, 1, 19, 1, 1, 19, 19, 1, 1, 19, 24, 19, 1, 1, 20, 25, 25, 20, 1, 1, 24, 70, 65, 70, 24, 1, 1, 25, 90, 71, 71, 90, 25, 1, 1, 65, 231, 230, 70, 230, 231, 65, 1, 1, 66, 295, 671, 211, 211, 671, 295, 66, 1, 1, 70, 684, 941, 671, 84, 671, 941, 684, 70, 1

Offset

0,5

Comments

Row sums are: 1, 2, 21, 40, 64, 92, 255, 374, 1124, 2488, 4818, ...

Links

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

Formula

T(n, k, q) = Sum_{j=0..10} q^j * floor(A174093(n, k)/2^j), for q = 3.

Example

Triangle begins as:
  1;
  1,  1;
  1, 19,   1;
  1, 19,  19,   1;
  1, 19,  24,  19,   1;
  1, 20,  25,  25,  20,   1;
  1, 24,  70,  65,  70,  24,   1;
  1, 25,  90,  71,  71,  90,  25,   1;
  1, 65, 231, 230,  70, 230, 231,  65,   1;
  1, 66, 295, 671, 211, 211, 671, 295,  66,  1;
  1, 70, 684, 941, 671,  84, 671, 941, 684, 70, 1;

Mathematica

A174093[n_, k_]:= If[n<2, 1, Binomial[n-k+1, k] + Binomial[k+1, n-k]];
T[n_, k_, q_]:= Sum[q^j*Floor[A174093[n, k]/2^j], {j, 0, 10}];
Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 10 2021 *)

Prog

(Sage)
def A174093(n, k): return 1 if n<2 else binomial(n-k+1, k) + binomial(k+1, n-k)
def T(n, k, q): return sum( q^j*(A174093(n, k)//2^j) for j in (0..10) )
flatten([[T(n, k, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 10 2021
(Magma)
A174093:= func< n, k | n lt 2 select 1 else Binomial(n-k+1, k) + Binomial(k+1, n-k) >;
T:= func< n, k, q | (&+[ q^j*Floor(A174093(n, k)/2^j): j in [0..10]]) >;
[T(n, k, 3): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 10 2021

Crossrefs

Cf. A174093 (q=0), A174095 (q=1), A174096 (q=2), this sequence (q=3).

Sequence in context: A040363 A040362 A040361 * A174040 A176078 A022182

Adjacent sequences: A174094 A174095 A174096 * A174098 A174099 A174100

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Mar 07 2010

Extensions

Edited by G. C. Greubel, Feb 10 2021

Status

approved