A219120 Triangle, read by rows, where T(n,k) is defined for n>=1, k=1..2*n-1, by a formula analogous to the second-order Eulerian triangle A008517.

1, 1, 1, -1, 1, 5, -2, -2, 1, 1, 15, 13, -19, 3, 3, -1, 1, 37, 128, -26, -74, 46, -4, -4, 1, 1, 83, 679, 755, -654, -68, 230, -90, 5, 5, -1, 1, 177, 2866, 9048, 2091, -5741, 1856, 498, -545, 155, -6, -6, 1, 1, 367, 10721, 67541, 98069, -24675, -35027, 22717, -3773, -1673, 1099, -245, 7, 7, -1, 1

Offset: 1

Author

Paul D. Hanna, Nov 12 2012

Keywords

• sign
• tabf

Comments

Compare to the o.g.f. of row n, E2(x,n), in the second-order Eulerian triangle A008517:

E2(x,n) = (1-x)^(2*n+1) * Sum_{k>=0} k^n * k^k * exp(-k*x) * x^k/k!.

Examples

			Triangle begins:
1;
1, 1, -1;
1, 5, -2, -2, 1;
1, 15, 13, -19, 3, 3, -1;
1, 37, 128, -26, -74, 46, -4, -4, 1;
1, 83, 679, 755, -654, -68, 230, -90, 5, 5, -1;
1, 177, 2866, 9048, 2091, -5741, 1856, 498, -545, 155, -6, -6, 1;
1, 367, 10721, 67541, 98069, -24675, -35027, 22717, -3773, -1673, 1099, -245, 7, 7, -1;
1, 749, 37300, 409170, 1290116, 863168, -590008, -108832, 182806, -65858, 5824, 4228, -1988, 364, -8, -8, 1; ...
		

Links

• Paul D. Hanna, Rows n = 1..32, flattened.

Crossrefs

Cf. A219121, A217900, A008517, A001147.

Programs

PARI

{T(n,k)=polcoeff((1-x)^(2*n-1)*sum(j=0,2*n,(j^n)*(j+1)^(j-1)*x^j/j!*exp(-(j+1)*x +O(x^k))),k)}
for(n=1,10,for(k=1,2*n-1,print1(T(n,k),", "));print(""))

Formula

O.g.f. of row n, R(x,n), is given by:

R(x,n) = (1-x)^(2*n-1) * Sum_{k>=0} k^n *(k+1)^(k-1) * exp(-(k+1)*x) * x^k/k!.

Row sums = A001147, which is the odd double factorials.

Column 1 = A050488(n-1), where A050488(n) = 3*(2^n-1) - 2*n.

Central terms of rows = A219121.