A174095 Triangle T(n,k,q) = Sum_{j=0..10} q^j * floor(A174093(n,k)/2^j) with q=1, read by rows.

1, 1, 1, 1, 7, 1, 1, 7, 7, 1, 1, 7, 10, 7, 1, 1, 8, 11, 11, 8, 1, 1, 10, 18, 15, 18, 10, 1, 1, 11, 26, 19, 19, 26, 11, 1, 1, 15, 39, 38, 18, 38, 39, 15, 1, 1, 16, 53, 67, 31, 31, 67, 53, 16, 1, 1, 18, 70, 109, 67, 22, 67, 109, 70, 18, 1

Offset

0,5

Comments

Row sums are: 1, 2, 9, 16, 26, 40, 73, 114, 204, 336, 552, ...

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 = 1.

Example

Triangle begins as:
  1;
  1,  1;
  1,  7,  1;
  1,  7,  7,   1;
  1,  7, 10,   7,  1;
  1,  8, 11,  11,  8,  1;
  1, 10, 18,  15, 18, 10,  1;
  1, 11, 26,  19, 19, 26, 11,   1;
  1, 15, 39,  38, 18, 38, 39,  15,  1;
  1, 16, 53,  67, 31, 31, 67,  53, 16,  1;
  1, 18, 70, 109, 67, 22, 67, 109, 70, 18, 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, 1], {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, 1) 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, 1): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 10 2021

Crossrefs

Cf. A174093 (q=0), this sequence (q=1), A174096 (q=2), A174097 (q=3).

Sequence in context: A120437 A336459 A367764 * A305607 A229779 A050179

Adjacent sequences: A174092 A174093 A174094 * A174096 A174097 A174098

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Mar 07 2010

Extensions

Edited by G. C. Greubel, Feb 10 2021

Status

approved