A174446 Triangle T(n, k, q) = ceiling(binomial(n, k)/f(n, q)) with T(0, 0) = 1, f(n, q) = 1 + tanh((n-1)/q), and q = 1, read by rows.

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 4, 3, 1, 1, 3, 6, 6, 3, 1, 1, 4, 8, 11, 8, 4, 1, 1, 4, 11, 18, 18, 11, 4, 1, 1, 5, 15, 29, 36, 29, 15, 5, 1, 1, 5, 19, 43, 64, 64, 43, 19, 5, 1, 1, 6, 23, 61, 106, 127, 106, 61, 23, 6, 1, 1, 6, 28, 83, 166, 232, 232, 166, 83, 28, 6, 1

Offset

0,5

Comments

For what pairs (n,q) is f(n,q) = 1 + floor(binomial(n,q)/2)? - Clark Kimberling, Jul 30 2011

Links

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

Formula

T(n, k, q) = ceiling(binomial(n, k)/f(n, q)) with T(0, 0) = 1, f(n, q) = 1 + tanh((n-1)/q), and q = 1.

Example

Triangle begins as:
  1;
  1, 1;
  1, 2,  1;
  1, 2,  2,  1;
  1, 3,  4,  3,   1;
  1, 3,  6,  6,   3,   1;
  1, 4,  8, 11,   8,   4,   1;
  1, 4, 11, 18,  18,  11,   4,  1;
  1, 5, 15, 29,  36,  29,  15,  5,  1;
  1, 5, 19, 43,  64,  64,  43, 19,  5, 1;
  1, 6, 23, 61, 106, 127, 106, 61, 23, 6, 1;

Mathematica

f[n_, q_]= 1 + Tanh[(n-1)/q];
T[n_, k_, q_]= If[n==0, 1, Ceiling[Binomial[n, k]/f[n, q]]];
Table[T[n, k, 1], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Magma)
T:= func< n, k, q | n eq 0 select 1 else Ceiling(Binomial(n, k)/(1 + Tanh((n-1)/q))) >;
[T(n, k, 1): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 05 2021
(Sage)
def T(n, k, q): return 1 if (n==0) else ceil(binomial(n, k)/(1 + tanh((n-1)/q)))
flatten([[T(n, k, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 05 2021

Crossrefs

Cf. this sequence (q=1), A174447 (q=4), A174448 (q=12).

Sequence in context: A048570 A090806 A241926 * A071201 A318045 A240656

Adjacent sequences: A174443 A174444 A174445 * A174447 A174448 A174449

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 20 2010

Extensions

Edited by G. C. Greubel, Aug 05 2021

Status

approved