A143596 A symmetrical triangle sequence of coefficients of a difference polynomial' p(x,n)=((x + 1)^(2*n) - (x^2 + 1)^n)/(2*x).

1, 2, 2, 2, 3, 6, 10, 6, 3, 4, 12, 28, 32, 28, 12, 4, 5, 20, 60, 100, 126, 100, 60, 20, 5, 6, 30, 110, 240, 396, 452, 396, 240, 110, 30, 6, 7, 42, 182, 490, 1001, 1484, 1716, 1484, 1001, 490, 182, 42, 7, 8, 56, 280, 896, 2184, 3976, 5720, 6400, 5720, 3976, 2184, 896, 280

Offset

1,2

Comments

Row sums are:{1, 6, 28, 120, 496, 2016, 8128, 32640, 130816, 523776}.

Links

Table of n, a(n) for n=1..62.

Formula

p(x,n)=((x + 1)^(2*n) - (x^2 + 1)^n)/(2*x); t(n,m)=coefficients(t(n,m)).

Example

{1},
{2, 2, 2},
{3, 6, 10, 6, 3},
{4, 12, 28, 32, 28, 12, 4},
{5, 20, 60, 100, 126, 100, 60, 20, 5},
{6, 30, 110, 240, 396, 452, 396, 240, 110, 30, 6},
{7, 42, 182, 490, 1001, 1484, 1716, 1484, 1001, 490, 182, 42, 7},
{8, 56, 280, 896, 2184, 3976, 5720, 6400, 5720, 3976, 2184, 896, 280, 56, 8},
{9, 72, 408, 1512, 4284, 9240, 15912, 21816, 24310, 21816, 15912, 9240, 4284, 1512, 408, 72, 9},
{10, 90, 570, 2400, 7752, 19320, 38760, 62880, 83980, 92252, 83980, 62880, 38760, 19320, 7752, 2400, 570, 90, 10}

Mathematica

Clear[p, x, n, m]; p[x_, n_] = ((x + 1)^(2*n) - (x^2 + 1)^n)/(2*x); Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}]; Flatten[%]

Crossrefs

Sequence in context: A367088 A321380 A218694 * A342763 A091712 A125721

Adjacent sequences: A143593 A143594 A143595 * A143597 A143598 A143599

Keyword

nonn,tabf

Author

Roger L. Bagula and Gary W. Adamson, Oct 25 2008

Status

approved