A211230 Matrix square of lower triangular array A211226.

1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 8, 8, 6, 4, 1, 8, 8, 8, 6, 2, 1, 20, 24, 24, 24, 9, 6, 1, 16, 20, 24, 24, 12, 9, 2, 1, 48, 64, 80, 96, 48, 48, 12, 8, 1, 32, 48, 64, 80, 48, 48, 16, 12, 2, 1, 112, 160, 240, 320, 200, 240, 80, 80, 15, 10, 1, 64, 112, 160, 240, 160

Offset: 0

Author

Peter Bala, Apr 05 2012

Keywords

• nonn
• easy
• tabl

Comments

Analog of square of Pascal's triangle.

Examples

			Triangle begins
.n\k.|....0....1....2....3....4....5....6....7....8....9
= = = = = = = = = = = = = = = = = = = = = = = = = = = = =
..0..|....1
..1..|....2....1
..2..|....3....2....1
..3..|....4....3....2....1
..4..|....8....8....6....4....1
..5..|....8....8....8....6....2....1
..6..|...20...24...24...24....9....6....1
..7..|...16...20...24...24...12....9....2....1
..8..|...48...64...80...96...48...48...12....8....1
..9..|...32...48...64...80...48...48...16...12....2....1
...
		

Crossrefs

Cf. A211226, A211231 (row sums).

Formula

T(2*n,2*k) = T(2*n+1,2*k+1) = (n+2-k)*binomial(n,k)*2^(n-k-1);

T(2*n,2*k+1) = n*binomial(n-1,k)*2^(n-k); T(2*n+1,2*k) = binomial(n,k)*2^(n-k+1).

Recurrence equations:

T(2*n,2*k) = n/k*T(2*n-1,2*k-1), T(2*n,2*k+1) = n*T(2*n-1,2*k);

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

O.g.f.: P(x,t)/Q(x,t), where P(x,t) = 1 + (x+2)*t - (1-x)^2*t^2 - (x^3+2*x^2+x+4)*t^3 and Q(x,t) = (1-(x^2+2)*t^2)^2.

Row polynomials:

R(2*n,x) = (x^2+2*n*x+n+2)*(x^2+2)^(n-1);

R(2*n+1,x) = (x^3+2*x^2+(n+2)*x+4)*(x^2+2)^(n-1).

Column 0 = A211227. Row sums A211231.