A139525 A trinomial made into a single variable polynomial coefficient triangle form three Pascallike factorial binomials in {x,y},{y,z} and {x,z}: binomials as: g(n,m)= If[m < "is less than" Floor[n/2], m!, (n - m)! ]; f(x,y,n)=Sum[g(n, i)*x^i*y^(n - i), {i, 0, n}]; p(x,y,z,n)=f(x,y,n)+f(y,z,n)+f(x,z,n).

3, 4, 2, 5, 2, 2, 7, 4, 2, 2, 8, 2, 4, 2, 2, 14, 2, 12, 4, 2, 2, 16, 2, 4, 12, 4, 2, 2, 40, 2, 4, 48, 12, 4, 2, 2, 46, 2, 4, 12, 48, 12, 4, 2, 2, 166, 2, 4, 12, 240, 48, 12, 4, 2, 2, 190, 2, 4, 12, 48, 240, 48, 12, 4, 2, 2

Offset

1,1

Comments

Row sums are:

{3, 6, 9, 15, 18, 36, 42, 114, 132, 492, 564}

References

Advanced Number Theory, Harvey Cohn, Dover Books, 1963, Pages 88-89

Links

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

Formula

g(n,m)= If[m < "is less than" Floor[n/2], m!, (n - m)! ]; f(x,y,n)=Sum[Binomial[n, i]*x^i*y^(n - i), {i, 0, n}]; p(x,y,z,n)=f(x,y,n)+f(y,z,n)+f(x,z,n); Out_n,m=Coefficients(p(x,1,1,n)).

Example

{3},
{4, 2},
{5, 2, 2},
{7, 4, 2, 2},
{8, 2, 4, 2, 2},
{14, 2, 12, 4, 2, 2},
{16, 2, 4, 12, 4, 2, 2},
{40, 2, 4, 48, 12, 4, 2, 2},
{46, 2, 4, 12, 48, 12, 4, 2, 2},
{166, 2, 4, 12, 240, 48, 12, 4, 2, 2},
{190, 2, 4, 12, 48, 240, 48, 12, 4, 2, 2}

Mathematica

Clear[f, x, n] g[n_, m_] = If[m < Floor[n/2], m!, (n - m)! ]; f[x_, y_, n_] = Sum[g[n, i]*x^i*y^(n - i), {i, 0, 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

Cf. A007283.

Sequence in context: A078069 A090131 A152833 * A246832 A133570 A117041

Adjacent sequences: A139522 A139523 A139524 * A139526 A139527 A139528

Keyword

nonn,uned

Author

Roger L. Bagula and Gary W. Adamson, Jun 09 2008

Status

approved