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A185414 Square array, read by antidiagonals, used to recursively calculate the zigzag numbers A000111. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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 * A133611 A010094 A019710

Adjacent sequences:  A185411 A185412 A185413 * A185415 A185416 A185417

KEYWORD

nonn,easy,tabl

AUTHOR

Peter Bala, Jan 26 2011

STATUS

approved

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Last modified February 20 01:26 EST 2018. Contains 299357 sequences. (Running on oeis4.)