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A185416 Square array, read by antidiagonals, used to recursively calculate A080635.

%I A185416 #10 Jan 07 2013 02:31:37
%S A185416 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,
%T A185416 27,6,1,7281,4194,1767,594,165,38,7,1,54351,30816,12699,4176,1143,264,
%U A185416 51,8,1,448821,251586,101979,32922,8865,2034,399,66,9,1
%N A185416 Square array, read by antidiagonals, used to recursively calculate A080635.
%C A185416 The table entries T(n,k), n,k>=1, are defined by the recurrence relation
%C A185416 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.
%C A185416 The first column of the table is A080635.
%C A185416 For similar tables to calculate the zigzag numbers, the Springer numbers and the number of minimax trees see A185414, A185418 and A185420, respectively.
%F A185416 (1)... T(n,k) = P(n,k)/k, where P(n,x) are the polynomials defined in A185415.
%e A185416 Triangle begins
%e A185416 n\k|....1......2......3......4......5.......6.......7
%e A185416 =====================================================
%e A185416 ..1|....1......1......1......1......1.......1.......1
%e A185416 ..2|....1......2......3......4......5.......6.......7
%e A185416 ..3|....3......6.....11.....18.....27......38......51
%e A185416 ..4|....9.....24.....51.....96....165.....264.....399
%e A185416 ..5|...39....114....279....594...1143....2034....3399
%e A185416 ..6|..189....648...1767...4176...8865...17304...31563
%e A185416 ..7|.1107...4194..12699..32922..76203..161442..318339
%e A185416 ..
%e A185416 Examples of the recurrence:
%e A185416 T(4,4) = 96 = 3*T(3,3)-4*T(3,4)+5*T(3,5) = 3*11-4*18+ 5*27;
%e A185416 T(5,1) = 39 = 0*T(4,0)-1*T(4,1)+2*T(4,2) = -1*9+2*24;
%p A185416 #A185416
%p A185416 P := proc(n,x) description 'polynomial sequence P(n,x) A185415'
%p A185416 if n = 0 return 1
%p A185416 else return
%p A185416 x*(P(n-1,x-1)-P(n-1,x)+P(n-1,x+1))
%p A185416 end proc:
%p A185416 for n from 1 to 10 do
%p A185416 seq(P(n,k)/k,k = 1..10);
%p A185416 end do;
%o A185416 (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))}
%Y A185416 Cf. A080635, A185414, A185415, A185418, A185420.
%K A185416 nonn,easy,tabl
%O A185416 1,4
%A A185416 _Peter Bala_, Jan 28 2011