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

1,4

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.

Links

Table of n, a(n) for n=1..55.

Formula

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

Example

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;

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;

Prog

(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))}

Crossrefs

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

Sequence in context: A152860 A002350 A109267 * A193918 A298804 A155788

Adjacent sequences: A185413 A185414 A185415 * A185417 A185418 A185419

Keyword

nonn,easy,tabl

Author

Peter Bala, Jan 28 2011

Status

approved