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
KEYWORD
nonn,tabl
AUTHOR
Roger L. Bagula, Mar 20 2010
EXTENSIONS
Edited by G. C. Greubel, Aug 05 2021
STATUS
approved