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A185420
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Square array, read by antidiagonals, used to recursively calculate the number of minimax trees A080795.
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6
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1, 4, 1, 20, 5, 1, 128, 32, 6, 1, 1024, 256, 46, 7, 1, 9856, 2464, 432, 62, 8, 1, 110720, 27680, 4784, 662, 80, 9, 1, 1421312, 355328, 60864, 8224, 952, 100, 10, 1, 20525056, 5131264, 873664, 116128, 13048, 1308, 122, 11, 1
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OFFSET
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1,2
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COMMENTS
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The table entries T(n,k), for n,k>=1, are defined by means of the recurrence relation
(1)... T(n+1,k) = (2*k+2)*T(n,k+1)-(k-1)*T(n,k-1),
with boundary condition T(1,k) = 1.
The first column of the table gives A080795.
For similarly defined tables used to calculate the zigzag numbers A000111 and the Springer numbers A001586 see A185414 and A185418, respectively.
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LINKS
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FORMULA
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(1)... T(n,k) = M(n,k)/k with M(n,x) the polynomials described in A185419.
(2)... First column: T(n,1) = A080795(n).
(3)... Second column: T(n,2) = (1/4)*A080795(n+1).
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EXAMPLE
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Square array begins
n\k|......1.......2.......3........4.......5.........6
======================================================
..1|......1.......1.......1........1........1........1
..2|......4.......5.......6........7........8........9
..3|.....20......32......46.......62.......80......100
..4|....128.....256.....432......662......952.....1308
..5|...1024....2464....4784.....8224....13048....19544
..6|...9856...27680...60864...116128...201632...327096
..7|.110720..355328..873664..1833728..3460640..6046720
..
Examples of recurrence relation:
T(4,3) = 432 = 8*T(3,4) - 2*T(3,2) = 8*62 - 2*32;
T(6,2) = 27680 = 6*T(5,3) - 1*T(5,1) = 6*4784 - 1*1024.
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MAPLE
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M := proc(n, x) option remember;
description 'minimax polynomials M(n, x)'
if n = 0
return 1
else return
x*(2*M(n-1, x+1)-M(n-1, x-1))
end proc:
for n from 1 to 10 do
seq(M(n, k)/k, k = 1..10);
end do;
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MATHEMATICA
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M[n_, x_] := M[n, x] = If[n == 0, 1, x (2 M[n - 1, x + 1] - M[n - 1, x - 1])];
T[n_, k_] := M[n, k]/k;
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PROG
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(PARI) {T(n, k)=if(n<1|k<1, 0, if(n==1, 1, (2*k+2)*T(n-1, k+1)-(k-1)*T(n-1, k-1)))}
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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