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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
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified June 25 14:43 EDT 2019. Contains 324352 sequences. (Running on oeis4.)