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A185414 Square array, read by antidiagonals, used to recursively calculate the zigzag numbers A000111.

%I A185414 #16 Aug 07 2017 18:13:02
%S A185414 1,1,1,2,2,1,5,5,3,1,16,16,10,4,1,61,61,39,17,5,1,272,272,176,80,26,6,
%T A185414 1,1385,1385,903,421,145,37,7,1,7936,7936,5200,2464,880,240,50,8,1,
%U A185414 50521,50521,33219,15917,5825,1661,371,65,9,1
%N A185414 Square array, read by antidiagonals, used to recursively calculate the zigzag numbers A000111.
%C A185414 The table entries T(n,k), for n,k>=1, are defined by means of the recurrence relation (1)... T(n+1,k) = 1/2*{(k-1)*T(n,k-1)+(k+1)*T(n,k+1)}, with boundary condition T(1,k) = 1.
%C A185414 The first column of the table produces the sequence of zigzag numbers A000111. Cf. A185416, A185418 and A185420.
%C A185414 Diagonal T(n,n+1) = A290579(n) for n>=1. - _Paul D. Hanna_, Aug 07 2017
%F A185414 (1)... T(n,k) = Z(n,k)/k with Z(n,x) the zigzag polynomials described in A147309.
%e A185414 The array begins:
%e A185414 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
%e A185414 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...;
%e A185414 2, 5, 10, 17, 26, 37, 50, 65, 82, ...;
%e A185414 5, 16, 39, 80, 145, 240, 371, 544, 765, ...;
%e A185414 16, 61, 176, 421, 880, 1661, 2896, 4741, 7376, ...;
%e A185414 61, 272, 903, 2464, 5825, 12336, 23947, 43328, 73989, ...;
%e A185414 272, 1385, 5200, 15917, 41936, 98377, 210320, 416765, ...;
%e A185414 1385, 7936, 33219, 112640, 326965, 840960, 1962191, ...; ...
%e A185414 Examples of the recurrence:
%e A185414 T(4,4) = 80 = (3*T(3,3) + 5*T(3,5))/2 = (3*10 + 5*26)/2;
%e A185414 T(5,3) = 176 = (2*T(4,2) + 4*T(4,4))/2 = (2*16 + 4*80)/2;
%e A185414 T(6,2) = 272 = (1*T(5,1) + 3*T(5,3))/2 = (1*16 + 3*176)/2.
%p A185414 #A185414 Z := proc(n,x)
%p A185414 description 'zigzag polynomials A147309'
%p A185414 if n = 0 return 1 else return 1/2*x*(Z(n-1,x-1)+Z(n-1,x+1))
%p A185414 end proc:
%p A185414 # values of Z(n,x)/x
%p A185414 for n from 1 to 10 do seq(Z(n,k)/k, k = 1..10);
%p A185414 end do;
%o A185414 (PARI) {T(n,k)=if(n==1,1,((k-1)*T(n-1,k-1)+(k+1)*T(n-1,k+1))/2)}
%o A185414 for(n=1,10, for(k=1,10, print1(T(n,k),", ")); print(""))
%Y A185414 Cf. A000111, A147309, A185416, A185418, A185420, A290579.
%K A185414 nonn,easy,tabl
%O A185414 1,4
%A A185414 _Peter Bala_, Jan 26 2011