A139585 A triangle of coefficients from truncated Pascal binomials in {x,y},y,z}and {z,x} in which the x^n,y^n and z^n terms are deleted: f(x,y,n)=(x+y)^n-(x^n+y^n); p(x,y,z,n)=f(x,y,n)+f(y,z,n)+f(z,x,n).

-3, 0, 2, 4, 6, 6, 6, 14, 8, 12, 8, 30, 10, 20, 20, 10, 62, 12, 30, 40, 30, 12, 126, 14, 42, 70, 70, 42, 14, 254, 16, 56, 112, 140, 112, 56, 16, 510, 18, 72, 168, 252, 252, 168, 72, 18, 1022, 20, 90, 240, 420, 504, 420, 240, 90, 20

Offset

1,1

Comments

Row sums are:

{-3, 0, 6, 18, 42, 90, 186, 378, 762, 1530, 3066};

These polynomials are inspired by the looping hyperfolium projections of the type:

f[x,i,n)=Binomial[n,i]*x^i/(1+x^n);

in n-1 coordinates.

Links

Table of n, a(n) for n=1..56.

Formula

f(x,y,n)=(x+y)^n-(x^n+y^n); p(x,y,z,n)=f(x,y,n)+f(y,z,n)+f(z,x,n); Out_n,m=Coefficients(p(x,1,1,n).

Example

Triangle begins:
  {-3},
  {0},
  {2, 4},
  {6, 6, 6},
  {14, 8, 12, 8},
  {30, 10, 20, 20, 10},
  {62, 12, 30, 40, 30, 12},
  {126, 14, 42, 70, 70, 42, 14},
  {254, 16, 56, 112, 140, 112, 56, 16},
  {510, 18, 72, 168, 252, 252, 168, 72, 18},
  {1022, 20, 90, 240, 420, 504, 420, 240, 90, 20}

Mathematica

f[x_, y_, n_] := (x + y)^n - x^n - y^n Table[ExpandAll[f[x, y, n] + f[y, z, n] + f[x, z, n]], {n, 0, 10}]; a = Table[CoefficientList[ExpandAll[f[x, y, n] + f[y, z, n] + f[x, z, n]] /. y -> 1 /. z -> 1, x], {n, 0, 10}]; Flatten[a]

Crossrefs

Sequence in context: A092154 A177344 A393336 * A273084 A349728 A261163

Adjacent sequences: A139582 A139583 A139584 * A139586 A139587 A139588

Keyword

uned,sign,tabf

Author

Roger L. Bagula and Gary W. Adamson, Jun 11 2008

Status

approved