|
| |
|
|
A167584
|
|
The ED4 array read by antidiagonals
|
|
14
|
|
|
|
1, 2, 1, 13, 6, 1, 76, 41, 10, 1, 789, 372, 93, 14, 1, 7734, 4077, 1020, 169, 18, 1, 110937, 53106, 13269, 2212, 269, 22, 1, 1528920, 795645, 198990, 33165, 4140, 393, 26, 1, 28018665, 13536360, 3383145, 563850, 70485, 6996, 541, 30, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
The coefficients in the upper right triangle of the ED4 array (m>n) were found with the a(n,m) formula while the coefficients in the lower left triangle of the ED4 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 ED3 arrays see A167546, A167560 and A167572.
|
|
|
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-3)!!/(2*(2*m-2*n-3)!!))*int(sinh(y*(2*n))/ (cosh(y))^(2*m-1),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!
|
|
|
EXAMPLE
|
The ED4 array begins with:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1
2, 6, 10, 14, 18, 22, 26, 30, 34, 38
13, 41, 93, 169, 269, 393, 541, 713, 909, 1129
76, 372, 1020, 2212, 4140, 6996, 10972, 16260, 23052, 31540
789, 4077, 13269, 33165, 70485, 133869, 233877, 382989, 595605, 888045
7734, 53106, 198990, 563850, 1339110, 2812194, 5389566, 9619770, 16216470, 26081490
|
|
|
CROSSREFS
|
A000012, A016825, A167585, A167586 and A167587 equal the first five rows of the array.
A024199, A167588 and A167589 equal the first three columns of the array.
A167590 equals the row sums of the ED4 array read by antidiagonals.
A167591 is a triangle related to the a(n) formulae of the rows of the ED4 array.
A167594 is a triangle related to the GF(z) formulae of the rows of the ED4 array.
Cf. A002866 (the 2^(n-1)*n! factor).
Cf. A167546 (ED1 array), A167560 (ED2 array), A167572 (ED3 array).
Sequence in context: A074955 A143663 A064079 * A112226 A192795 A074808
Adjacent sequences: A167581 A167582 A167583 * A167585 A167586 A167587
|
|
|
KEYWORD
|
nonn,tabl
|
|
|
AUTHOR
|
Johannes W. Meijer, Nov 10 2009
|
|
|
STATUS
|
approved
|
| |
|
|
