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

Author

Roger L. Bagula, Mar 07 2010

Keywords

• nonn
• tabl
• easy
• less

Comments

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

Examples

			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;
		

Links

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

Crossrefs

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

Programs

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

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 *)

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

Formula

T(n, k, q) = Sum_{j=0..10} q^j * floor(A174093(n, k)/2^j), for q = 1.

Extensions

Edited by G. C. Greubel, Feb 10 2021