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

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, 1, 1385, 1385, 903, 421, 145, 37, 7, 1, 7936, 7936, 5200, 2464, 880, 240, 50, 8, 1, 50521, 50521, 33219, 15917, 5825, 1661, 371, 65, 9, 1

Offset: 1

Author

Peter Bala, Jan 26 2011

Keywords

• nonn
• easy
• tabl

Comments

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.

The first column of the table produces the sequence of zigzag numbers A000111. Cf. A185416, A185418 and A185420.

Diagonal T(n,n+1) = A290579(n) for n>=1. - Paul D. Hanna, Aug 07 2017

Examples

			The array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...;
2, 5, 10, 17, 26, 37, 50, 65, 82, ...;
5, 16, 39, 80, 145, 240, 371, 544, 765, ...;
16, 61, 176, 421, 880, 1661, 2896, 4741, 7376, ...;
61, 272, 903, 2464, 5825, 12336, 23947, 43328, 73989, ...;
272, 1385, 5200, 15917, 41936, 98377, 210320, 416765, ...;
1385, 7936, 33219, 112640, 326965, 840960, 1962191, ...; ...
Examples of the recurrence:
T(4,4) = 80 = (3*T(3,3) + 5*T(3,5))/2 = (3*10 + 5*26)/2;
T(5,3) = 176 = (2*T(4,2) + 4*T(4,4))/2 = (2*16 + 4*80)/2;
T(6,2) = 272 = (1*T(5,1) + 3*T(5,3))/2 = (1*16 + 3*176)/2.
		

Crossrefs

Cf. A000111, A147309, A185416, A185418, A185420, A290579.

Programs

Maple

#A185414 Z := proc(n,x)
description 'zigzag polynomials A147309'
if n = 0 return 1 else return 1/2*x*(Z(n-1,x-1)+Z(n-1,x+1))
end proc:
# values of Z(n,x)/x
for n from 1 to 10 do seq(Z(n,k)/k, k = 1..10);
end do;

PARI

{T(n,k)=if(n==1,1,((k-1)*T(n-1,k-1)+(k+1)*T(n-1,k+1))/2)}
for(n=1,10, for(k=1,10, print1(T(n,k),", ")); print(""))

Formula

(1)... T(n,k) = Z(n,k)/k with Z(n,x) the zigzag polynomials described in A147309.