A143596 Triangle read by rows: T(n,k) is the coefficient of x^k in p(x,n) = ((x + 1)^(2*n) - (x^2 + 1)^n)/(2*x), 0 <= k <= 2*n-1.

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

Author

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

Keywords

• nonn
• tabf

Comments

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

Examples

			{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},
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..2500 (first 50 rows)

Crossrefs

Cf. A006516 (row sums).

Programs

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[%]

PARI

row(n)=Vecrev(((x + 1)^(2*n) - (x^2 + 1)^n)/(2*x)) \\ Andrew Howroyd, Sep 06 2024

Formula

T(n,k) = [x^k] ((x + 1)^(2*n) - (x^2 + 1)^n)/(2*x).

T(n,k) = T(n,2*n-1-k).

G.f.: x/(1 - 2*(1 + y + y^2)*x + (1 + y)^2*(1 + y^2)*x^2). - Andrew Howroyd, Sep 06 2024