%I #7 Nov 08 2018 18:06:31
%S 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,
%T 70,10,2,1,1,0,272,1952,656,75,10,2,1,1,0,1385,17408,7442,816,75,10,2,
%U 1,1,0,7936,177280,99280,11407,832,75,10,2,1,1
%N 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.
%C See the comments and definitions in A320956. Note also the corresponding construction for the exp function in A320955.
%e Array starts:
%e n\k 0 1 2 3 4 5 6 7 8 ...
%e -------------------------------------------------------
%e [0] 1, 0, 0, 0, 0, 0, 0, 0, 0, ... A000007
%e [1] 1, 1, 1, 2, 5, 16, 61, 272, 1385, ... A000111
%e [2] 1, 1, 2, 8, 40, 256, 1952, 17408, 177280, ... A000828
%e [3] 1, 1, 2, 10, 70, 656, 7442, 99280, 1515190, ... A320957
%e [4] 1, 1, 2, 10, 75, 816, 11407, 194480, 3871075, ... A321394
%e [5] 1, 1, 2, 10, 75, 832, 12322, 232560, 5325325, ...
%e [6] 1, 1, 2, 10, 75, 832, 12383, 238272, 5693735, ...
%e [7] 1, 1, 2, 10, 75, 832, 12383, 238544, 5732515, ...
%e [8] 1, 1, 2, 10, 75, 832, 12383, 238544, 5733900, ...
%e -------------------------------------------------------
%e Seen as a triangle given by descending antidiagonals:
%e [0] 1
%e [1] 0, 1
%e [2] 0, 1, 1
%e [3] 0, 1, 1, 1
%e [4] 0, 2, 2, 1, 1
%e [5] 0, 5, 8, 2, 1, 1
%e [6] 0, 16, 40, 10, 2, 1, 1
%e [7] 0, 61, 256, 70, 10, 2, 1, 1
%p sf := proc(n) option remember; `if`(n <= 1, 1-n, (n-1)*(sf(n-1) + sf(n-2))) end:
%p kernel := proc(n, k) option remember; binomial(n, k)*sf(k) end:
%p egf := n -> add(kernel(n, k)*((tan + sec)(x*(n - k))), k=0..n):
%p A321400Row := proc(n, len) series(egf(n), x, len + 2):
%p seq(coeff(%, x, k)*k!/n!, k=0..len) end:
%p seq(lprint(A321400Row(n, 9)), n=0..9);
%Y Cf. A000111 (n=1), A000828 (n=2), A320957 (n=3), A321394 (n=4), A320956 (limit).
%Y Antidiagonal sums (and row sums of the triangle): A321399.
%K nonn,tabl
%O 0,12
%A _Peter Luschny_, Nov 08 2018