A142469 The nonzero difference between the Pascal {1,8,1} level triangle sequence and the {1,8,1} Catalan generalized triangle: t0(n,m)=Binomial[n, m]*Product[k!*(n + k)!/((m + k)!*(n - m + k)!), {k, 1, 7}]. A(n,k)=(3*n - 3*k + 1)A(n - 1, k - 1) + (3*k - 2)A(n - 1, k); t(n,m)=A(n,m)-t0(n,m). The first three levels and the external columns are zero and extracted.

3, 3, 46, 6, 46, 347, 532, 532, 347, 1932, 14505, 740, 14505, 1932, 9199, 203925, 152405, 152405, 203925, 9199, 40250, 2087884, 6882086, -86372, 6882086, 2087884, 40250, 168318, 17968725, 152844537, 78623775, 78623775, 152844537, 17968725, 168318

Offset

1,1

Comments

Row sums are:

{6, 98, 1758, 33614, 731058, 17934068, 499210710}.

Links

Table of n, a(n) for n=1..35.

Formula

t0(n,m)=Binomial[n, m]*Product[k!*(n + k)!/((m + k)!*(n - m + k)!), {k, 1, 7}]. A(n,k)=(3*n - 3*k + 1)A(n - 1, k - 1) + (3*k - 2)A(n - 1, k); t(n,m)=A(n,m)-t0(n,m).

Example

{3, 3},
{46, 6, 46},
{347, 532, 532, 347},
{1932, 14505, 740, 14505, 1932},
{9199, 203925, 152405, 152405, 203925, 9199},
{40250, 2087884, 6882086, -86372, 6882086, 2087884, 40250},
{168318, 17968725, 152844537, 78623775, 78623775, 152844537, 17968725, 168318}

Mathematica

f[0, 0] = 1; f[n_, m_] := f[n, m] = Binomial[n, m]*Product[k!*(n + k)!/((m + k)!*(n - m + k)!), {k, 1, 6}]; A[n_, 1] := 1 A[n_, n_] := 1 A[n_, k_] := (3*n - 3*k + 1)A[n - 1, k - 1] + (3*k - 2)A[n - 1, k]; a = Table[If[f[n - 1, k - 1] - A[n, k] == 0, {}, -f[n - 1, k - 1] + A[n, k]], {n, 10}, {k, n}]; c = Delete[Union[Table[Flatten[a[[n]]], {n, 1, 10}]], 1]; Flatten[a]

Crossrefs

Sequence in context: A290366 A083391 A165636 * A009809 A265717 A292753

Adjacent sequences: A142466 A142467 A142468 * A142470 A142471 A142472

Keyword

sign,uned

Author

Roger L. Bagula and Gary W. Adamson, Sep 20 2008

Status

approved