A185416 Square array, read by antidiagonals, used to recursively calculate A080635.

1, 1, 1, 3, 2, 1, 9, 6, 3, 1, 39, 24, 11, 4, 1, 189, 114, 51, 18, 5, 1, 1107, 648, 279, 96, 27, 6, 1, 7281, 4194, 1767, 594, 165, 38, 7, 1, 54351, 30816, 12699, 4176, 1143, 264, 51, 8, 1, 448821, 251586, 101979, 32922, 8865, 2034, 399, 66, 9, 1

Offset: 1

Author

Peter Bala, Jan 28 2011

Keywords

• nonn
• easy
• tabl

Comments

The table entries T(n,k), n,k>=1, are defined by the recurrence relation

1)... T(n+1,k) = (k-1)*T(n,k-1)-k*T(n,k)+(k+1)*T(n,k+1) with boundary condition T(1,k)=1.

The first column of the table is A080635.

For similar tables to calculate the zigzag numbers, the Springer numbers and the number of minimax trees see A185414, A185418 and A185420, respectively.

Examples

			Triangle begins
n\k|....1......2......3......4......5.......6.......7
=====================================================
..1|....1......1......1......1......1.......1.......1
..2|....1......2......3......4......5.......6.......7
..3|....3......6.....11.....18.....27......38......51
..4|....9.....24.....51.....96....165.....264.....399
..5|...39....114....279....594...1143....2034....3399
..6|..189....648...1767...4176...8865...17304...31563
..7|.1107...4194..12699..32922..76203..161442..318339
..
Examples of the recurrence:
T(4,4) = 96 = 3*T(3,3)-4*T(3,4)+5*T(3,5) = 3*11-4*18+ 5*27;
T(5,1) = 39 = 0*T(4,0)-1*T(4,1)+2*T(4,2) = -1*9+2*24;
		

Crossrefs

Cf. A080635, A185414, A185415, A185418, A185420.

Programs

Maple

#A185416
P := proc(n,x) description 'polynomial sequence P(n,x) A185415'
if n = 0 return 1
else return
x*(P(n-1,x-1)-P(n-1,x)+P(n-1,x+1))
end proc:
for n from 1 to 10 do
seq(P(n,k)/k,k = 1..10);
end do;

PARI

{T(n, k)=if(n==1, 1, (k-1)*T(n-1, k-1)-k*T(n-1,k)+(k+1)*T(n-1, k+1))}

Formula

(1)... T(n,k) = P(n,k)/k, where P(n,x) are the polynomials defined in A185415.