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A321400
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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.
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1
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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
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OFFSET
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0,12
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COMMENTS
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See the comments and definitions in A320956. Note also the corresponding construction for the exp function in A320955.
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LINKS
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EXAMPLE
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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
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MAPLE
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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);
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CROSSREFS
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Antidiagonal sums (and row sums of the triangle): A321399.
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KEYWORD
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AUTHOR
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STATUS
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approved
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