A185420 Square array, read by antidiagonals, used to recursively calculate the number of minimax trees A080795.

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

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

Table of n, a(n) for n=1..45.

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

#A185420
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

Cf. A080795, A185414, A185416, A185418, A185419.

Sequence in context: A144885 A144886 A117380 * A167432 A201639 A078939

Adjacent sequences: A185417 A185418 A185419 * A185421 A185422 A185423

Keyword

nonn,easy,tabl,changed

Author

Peter Bala, Jan 30 2011

Status

approved