A167572 The ED3 array read by antidiagonals

1, 5, 1, 23, 11, 1, 167, 83, 17, 1, 1473, 741, 183, 23, 1, 16413, 8169, 2043, 323, 29, 1, 211479, 106107, 26529, 4409, 503, 35, 1, 3192975, 1592235, 398025, 66345, 8175, 723, 41, 1, 54010305, 27062325, 6765975, 1127655, 140865, 13677, 983, 47, 1

Offset

1,2

Comments

The coefficients in the upper right triangle of the ED3 array (m>n) were found with the a(n,m) formula while the coefficients in the lower left triangle of the ED3 array (m<=n) were found with the recurrence relation, see below. We use for the array rows the letter n (>=1) and for the array columns the letter m (>=1).

For the ED1, ED2 and ED4 arrays see A167546, A167560 and A167584.

Links

Table of n, a(n) for n=1..45.

Johannes W. Meijer, The four Escher-Droste arrays, jpg image, Mar 08 2013.

Formula

a(n,m) = ((2*m-1)!!/ (2*m-2*n-1)!!)*int(sinh(y*(2*n-1))/(cosh(y))^(2*m),y=0..infinity) for m>n.

The (n-1)-differences of the n-th array row lead to the recurrence relation

sum((-1)^k*binomial(n-1,k)*a(n,m-k),k=0..n-1) = 2^(n-1)*(n-1)!*(2*n-1).

Example

The ED3 array begins with:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1
5, 11, 17, 23, 29, 35, 41, 47, 53, 59
23, 83, 183, 323, 503, 723, 983, 1283, 1623, 2003
167, 741, 2043, 4409, 8175, 13677, 21251, 31233, 43959, 59765
1473, 8169, 26529, 66345, 140865, 266793, 464289, 756969, 1171905, 1739625
16413, 106107, 398025, 1127655, 2678325, 5623443, 10768737, 19194495, 32297805, 51834795

Crossrefs

A000012, A016969, A167573, A167574 and A167575 equal the first five rows of the array.

A167576, A167577 and A167578 equal the first three columns of the array.

A167579 equals the row sums of the ED3 array read by antidiagonals.

A167580 is a triangle related to the a(n) formulas of the rows of the ED3 array.

A167583 is a triangle related to the GF(z) formulas of the rows of the ED3 array.

Cf. A014481 (the 2^(n-1)*(n-1)!*(2*n-1) factor).

Cf. A167546 (ED1 array), A167560 (ED2 array), A167584 (ED4 array).

Sequence in context: A213118 A259682 A286897 * A147476 A146675 A201884

Adjacent sequences: A167569 A167570 A167571 * A167573 A167574 A167575

Keyword

nonn,tabl

Author

Johannes W. Meijer, Nov 10 2009

Status

approved