|
|
A174447
|
|
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 = 4, read by rows.
|
|
3
|
|
|
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 4, 3, 1, 1, 3, 6, 6, 3, 1, 1, 4, 9, 11, 9, 4, 1, 1, 4, 12, 19, 19, 12, 4, 1, 1, 5, 15, 29, 37, 29, 15, 5, 1, 1, 5, 19, 43, 65, 65, 43, 19, 5, 1, 1, 6, 23, 61, 107, 128, 107, 61, 23, 6, 1, 1, 6, 28, 84, 167, 233, 233, 167, 84, 28, 6, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,5
|
|
LINKS
|
|
|
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 = 4.
|
|
EXAMPLE
|
Triangle begins as:
1;
1, 1;
1, 2, 1;
1, 3, 3, 1;
1, 3, 4, 3, 1;
1, 3, 6, 6, 3, 1;
1, 4, 9, 11, 9, 4, 1;
1, 4, 12, 19, 19, 12, 4, 1;
1, 5, 15, 29, 37, 29, 15, 5, 1;
1, 5, 19, 43, 65, 65, 43, 19, 5, 1;
1, 6, 23, 61, 107, 128, 107, 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, 4], {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, 4): 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, 4) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 05 2021
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|