A211226 Triangular array: T(n,k) = f(n)/(f(k)*f(n-k)), where f(n) = (floor(n/2))!.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 6, 3, 3, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 4, 4, 12, 6, 12, 4, 4, 1, 1, 1, 4, 4, 6, 6, 4, 4, 1, 1, 1, 5, 5, 20, 10, 30, 10, 20, 5, 5, 1, 1, 1, 5, 5, 10, 10, 10, 10, 5, 5, 1, 1, 1, 6, 6, 30, 15

Offset: 0

Author

Peter Bala, Apr 05 2012

Keywords

• nonn
• easy
• tabl

Examples

			Triangle begins
.n\k.|....0....1....2....3....4....5....6....7....8....9...10...11
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
..0..|....1
..1..|....1....1
..2..|....1....1....1
..3..|....1....1....1....1
..4..|....1....2....2....2....1
..5..|....1....1....2....2....1....1
..6..|....1....3....3....6....3....3....1
..7..|....1....1....3....3....3....3....1....1
..8..|....1....4....4...12....6...12....4....4....1
..9..|....1....1....4....4....6....6....4....4....1....1
.10..|....1....5....5...20...10...30...10...20....5....5....1
.11..|....1....1....5....5...10...10...10...10....5....5....1....1
...
		

Links

• Peter Bala, Notes on A211226

Crossrefs

Cf. A007318, A056040, A211227 (row sums), A211228 (shallow diagonal sums), A211229 (inverse), A211230 (array squared).

Formula

T(n,k) := f(n)/(f(k)*f(n-k)), where f(n) := (floor(n/2))!.

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

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

Recurrence equations:

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

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

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

The Star of David property holds:

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

O.g.f.: (1 + t*(1+x) - t^2*(1-x+x^2) - t^3*(1+x+x^2+x^3))/(1 - t^2*(1+x^2))^2 = sum {n>=0} R(n,x)*t^n = 1 + (1+x)*t + (1+x+x^2)*t^2 + (1+x+x^2+x^3)*t^3 + ....

E.g.f.: cosh(t*sqrt(1+x^2)) + (1+x+x*t/2)/sqrt(1+x^2)*sinh(t*sqrt(1+x^2)) = sum {n>=0} R(n,x)*t^n/n! = 1 + (1+x)*t + (1+x+x^2)*t^2/2! + (1+x+x^2+x^3)*t^3/3! + ....

Row generating polynomials: R(2*n+1,x) = (1+x)*(1+x^2)^n; R(2*n,x) = (1+n*x+x^2)*(1+x^2)^(n-1).

Row sums: A211227. Shallow diagonal sums A211228. Central terms T(2*n,n) equal A056040(n).

The inverse array A211229 involves the derangement numbers A000166. The squared array is A211230.