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

1,4

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

Links

Table of n, a(n) for n=1..55.

Formula

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

Example

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.

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;

Prog

(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(""))

Crossrefs

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

Sequence in context: A134379 A108087 A123158 * A346520 A362925 A133611

Adjacent sequences: A185411 A185412 A185413 * A185415 A185416 A185417

Keyword

nonn,easy,tabl

Author

Peter Bala, Jan 26 2011

Status

approved