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A185420 Square array, read by antidiagonals, used to recursively calculate the number of minimax trees A080795. 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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.
See also A185416.
LINKS
FORMULA
(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).
EXAMPLE
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.
MAPLE
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;
MATHEMATICA
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;
Table[T[d - k + 1, k], {d, 1, 9}, {k, 1, d}] // Flatten (* Jean-François Alcover, Sep 24 2022 *)
PROG
(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)))}
CROSSREFS
Sequence in context: A144885 A144886 A117380 * A167432 A201639 A078939
KEYWORD
nonn,easy,tabl
AUTHOR
Peter Bala, Jan 30 2011
STATUS
approved

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Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)