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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 (list; table; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A152860 A002350 A109267 * A193918 A298804 A155788
KEYWORD
nonn,easy,tabl
AUTHOR
Peter Bala, Jan 28 2011
STATUS
approved

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Last modified April 14 18:28 EDT 2024. Contains 371667 sequences. (Running on oeis4.)