A321400 A family of sequences converging to the exponential limit of sec + tan (A320956). Square array A(n, k) for n >= 0 and k >= 0, read by descending antidiagonals.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 1, 1, 0, 5, 8, 2, 1, 1, 0, 16, 40, 10, 2, 1, 1, 0, 61, 256, 70, 10, 2, 1, 1, 0, 272, 1952, 656, 75, 10, 2, 1, 1, 0, 1385, 17408, 7442, 816, 75, 10, 2, 1, 1, 0, 7936, 177280, 99280, 11407, 832, 75, 10, 2, 1, 1

Offset

0,12

Comments

See the comments and definitions in A320956. Note also the corresponding construction for the exp function in A320955.

Links

Table of n, a(n) for n=0..65.

Example

Array starts:
n\k   0  1  2   3   4    5      6       7        8  ...
-------------------------------------------------------
  [0] 1, 0, 0,  0,  0,   0,     0,      0,       0, ... A000007
  [1] 1, 1, 1,  2,  5,  16,    61,    272,    1385, ... A000111
  [2] 1, 1, 2,  8, 40, 256,  1952,  17408,  177280, ... A000828
  [3] 1, 1, 2, 10, 70, 656,  7442,  99280, 1515190, ... A320957
  [4] 1, 1, 2, 10, 75, 816, 11407, 194480, 3871075, ... A321394
  [5] 1, 1, 2, 10, 75, 832, 12322, 232560, 5325325, ...
  [6] 1, 1, 2, 10, 75, 832, 12383, 238272, 5693735, ...
  [7] 1, 1, 2, 10, 75, 832, 12383, 238544, 5732515, ...
  [8] 1, 1, 2, 10, 75, 832, 12383, 238544, 5733900, ...
-------------------------------------------------------
Seen as a triangle given by descending antidiagonals:
  [0] 1
  [1] 0,  1
  [2] 0,  1,   1
  [3] 0,  1,   1,  1
  [4] 0,  2,   2,  1,  1
  [5] 0,  5,   8,  2,  1, 1
  [6] 0, 16,  40, 10,  2, 1, 1
  [7] 0, 61, 256, 70, 10, 2, 1, 1

Maple

sf := proc(n) option remember; `if`(n <= 1, 1-n, (n-1)*(sf(n-1) + sf(n-2))) end:
kernel := proc(n, k) option remember; binomial(n, k)*sf(k) end:
egf := n -> add(kernel(n, k)*((tan + sec)(x*(n - k))), k=0..n):
A321400Row := proc(n, len) series(egf(n), x, len + 2):
seq(coeff(%, x, k)*k!/n!, k=0..len) end:
seq(lprint(A321400Row(n, 9)), n=0..9);

Crossrefs

Cf. A000111 (n=1), A000828 (n=2), A320957 (n=3), A321394 (n=4), A320956 (limit).

Antidiagonal sums (and row sums of the triangle): A321399.

Sequence in context: A004564 A300705 A144157 * A004562 A261817 A123550

Adjacent sequences: A321397 A321398 A321399 * A321401 A321402 A321403

Keyword

nonn,tabl

Author

Peter Luschny, Nov 08 2018

Status

approved