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A185416
Square array, read by antidiagonals, used to recursively calculate A080635.
6
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.
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
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
KEYWORD
nonn,easy,tabl
AUTHOR
Peter Bala, Jan 28 2011
STATUS
approved