A136267 Alternate terms of A001263 as polynomials divided by x+1 to give a new triangle of coefficients of even powered polynomials.

1, 1, 5, 1, 1, 14, 36, 14, 1, 1, 27, 169, 321, 169, 27, 1, 1, 44, 496, 2024, 3268, 2024, 496, 44, 1, 1, 65, 1145, 7930, 24740, 36244, 24740, 7930, 1145, 65, 1, 1, 90, 2276, 23750, 119393, 310036, 426128, 310036, 119393, 23750, 2276, 90, 1, 1, 119, 4081, 59619

Offset

1,3

Comments

Row sums are:

Table[Apply[Plus, CoefficientList[Factor[a[[n]]]/(x + 1), x]], {n, 2, Length[a], 2}];

{1, 7, 66, 715, 8398, 104006, 1337220, 17678835, 238819350, 3282060210}.

This sequence was found while looking into Gary W. Adamson's comment on A001263.

Links

Table of n, a(n) for n=1..53.

Formula

T(n,m) = Binomial[n - 1, m - 1]*Binomial[n, m - 1]/m p(x,n)=Sum[t(n,m)^x^(m-1),{m,1,n}]/(x+1): {n,2,limit,skip one}

Example

{1},
{1, 5, 1},
{1, 14, 36, 14, 1},
{1, 27, 169, 321, 169, 27, 1},
{1, 44, 496, 2024, 3268, 2024, 496, 44, 1},
{1, 65, 1145, 7930, 24740, 36244, 24740, 7930, 1145, 65, 1},
{1, 90, 2276, 23750, 119393, 310036, 426128, 310036, 119393, 23750, 2276, 90,1},
{1, 119, 4081, 59619, 437241, 1748943, 3976777, 5225273, 3976777, 1748943,437241, 59619, 4081, 119, 1},
{1, 152, 6784, 131936, 1324624, 7511840, 25309312, 52054832, 66140388, 52054832, 25309312, 7511840, 1324624, 131936, 6784, 152, 1},
{1, 189, 10641, 265524, 3490320, 26556432, 123677328, 364582392, 693313668, 858267220, 693313668, 364582392, 123677328, 26556432, 3490320, 265524, 10641, 189, 1}

Mathematica

T[n_, m_] := Binomial[n - 1, m - 1]*Binomial[n, m - 1]/m; a = Table[Apply[Plus, Table[T[n, m]*x^(m - 1), {m, 1, n}]], {n, 1, 20}]; Table[Factor[a[[n]]]/(x + 1), {n, 2, Length[a], 2}]; b = Table[CoefficientList[Factor[a[[n]]]/(x + 1), x], {n, 2, Length[a], 2}]; Flatten[b]

Crossrefs

Cf. A001263.

Sequence in context: A144438 A157207 A008957 * A109960 A196019 A056940

Adjacent sequences: A136264 A136265 A136266 * A136268 A136269 A136270

Keyword

nonn,uned,tabf

Author

Roger L. Bagula and Gary W. Adamson, Mar 18 2008

Status

approved