A140284 A weighted binomial triangle with smooth weighting function: f(n,d) = Floor[1 + d*Sech[d/2 - n]].

1, 1, 1, 1, 6, 1, 1, 9, 9, 1, 1, 12, 30, 12, 1, 1, 15, 50, 50, 15, 1, 1, 12, 60, 140, 60, 12, 1, 1, 14, 63, 245, 245, 63, 14, 1, 1, 8, 84, 336, 630, 336, 84, 8, 1, 1, 9, 72, 336, 1008, 1008, 336, 72, 9, 1, 1, 10, 45, 360, 1470, 2772, 1470, 360, 45, 10, 1

Offset

1,5

Comments

Row sums are:

{1, 2, 4, 13, 46, 118, 275, 633, 1481, 2844, 6535}.;

Here the interior coefficients are larger than the

pascal triangle: most generalized Pascal triangles yield smaller interior

coefficients.

Links

Table of n, a(n) for n=1..66.

Formula

a(n,d)=If[n == 0 || n == d, 1, f[n, d]* Binomial[d, n]]; f(n,d) = Floor[1 + d*Sech[d/2 - n]].

Example

{{1},
{1, 1},
{1, 6, 1},
{1, 9, 9, 1},
{1, 12, 30, 12, 1},
{1, 15, 50, 50, 15,1},
{1, 12, 60, 140, 60, 12, 1},
{1, 14, 63, 245, 245, 63, 14, 1},
{1, 8, 84, 336, 630, 336, 84, 8, 1},
{1, 9, 72, 336, 1008, 1008, 336, 72, 9,1},
{1, 10, 45, 360, 1470, 2772, 1470, 360, 45, 10, 1}}

Mathematica

f[n_, d_] = Floor[1 + d*Sech[d/2 - n]]; a = Table[Table[If[n == 0 || n == d, 1, f[n, d]* Binomial[ d, n]], {n, 0, d}], {d, 0, 10}]; Flatten[a] Table[Apply[Plus, Table[If[n == 1 || n == d, 1, f[n, d]* Binomial[d, n]], {n, 0, d}]], {d, 0, 10}];

Crossrefs

Sequence in context: A011491 A189089 A132047 * A143087 A144470 A174377

Adjacent sequences: A140281 A140282 A140283 * A140285 A140286 A140287

Keyword

nonn,uned,tabl

Author

Roger L. Bagula and Gary W. Adamson, May 23 2008

Status

approved