cp's OEIS Frontend

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

%I A185420 #18 Jan 05 2025 23:40:21
%S A185420 1,4,1,20,5,1,128,32,6,1,1024,256,46,7,1,9856,2464,432,62,8,1,110720,
%T A185420 27680,4784,662,80,9,1,1421312,355328,60864,8224,952,100,10,1,
%U A185420 20525056,5131264,873664,116128,13048,1308,122,11,1
%N A185420 Square array, read by antidiagonals, used to recursively calculate the number of minimax trees A080795.
%C A185420 The table entries T(n,k), for n,k>=1, are defined by means of the recurrence relation
%C A185420 (1)... T(n+1,k) = (2*k+2)*T(n,k+1)-(k-1)*T(n,k-1),
%C A185420 with boundary condition T(1,k) = 1.
%C A185420 The first column of the table gives A080795.
%C A185420 For similarly defined tables used to calculate the zigzag numbers A000111 and the Springer numbers A001586 see A185414 and A185418, respectively.
%C A185420 See also A185416.
%F A185420 (1)... T(n,k) = M(n,k)/k with M(n,x) the polynomials described in A185419.
%F A185420 (2)... First column: T(n,1) = A080795(n).
%F A185420 (3)... Second column: T(n,2) = (1/4)*A080795(n+1).
%e A185420 Square array begins
%e A185420 n\k|......1.......2.......3........4.......5.........6
%e A185420 ======================================================
%e A185420 ..1|......1.......1.......1........1........1........1
%e A185420 ..2|......4.......5.......6........7........8........9
%e A185420 ..3|.....20......32......46.......62.......80......100
%e A185420 ..4|....128.....256.....432......662......952.....1308
%e A185420 ..5|...1024....2464....4784.....8224....13048....19544
%e A185420 ..6|...9856...27680...60864...116128...201632...327096
%e A185420 ..7|.110720..355328..873664..1833728..3460640..6046720
%e A185420 ..
%e A185420 Examples of recurrence relation:
%e A185420 T(4,3) = 432 = 8*T(3,4) - 2*T(3,2) = 8*62 - 2*32;
%e A185420 T(6,2) = 27680 = 6*T(5,3) - 1*T(5,1) = 6*4784 - 1*1024.
%p A185420 #A185420
%p A185420 M := proc(n,x) option remember;
%p A185420 description 'minimax polynomials M(n,x)'
%p A185420 if n = 0
%p A185420 return 1
%p A185420 else return
%p A185420 x*(2*M(n-1,x+1)-M(n-1,x-1))
%p A185420 end proc:
%p A185420 for n from 1 to 10 do
%p A185420 seq(M(n,k)/k, k = 1..10);
%p A185420 end do;
%t A185420 M[n_, x_] := M[n, x] = If[n == 0, 1, x (2 M[n - 1, x + 1] - M[n - 1, x - 1])];
%t A185420 T[n_, k_] := M[n, k]/k;
%t A185420 Table[T[d - k + 1, k], {d, 1, 9}, {k, 1, d}] // Flatten (* _Jean-François Alcover_, Sep 24 2022 *)
%o A185420 (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)))}
%Y A185420 Cf. A080795, A185414, A185416, A185418, A185419.
%K A185420 nonn,easy,tabl
%O A185420 1,2
%A A185420 _Peter Bala_, Jan 30 2011