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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.

%I A321400 #7 Nov 08 2018 18:06:31
%S A321400 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 A321400 70,10,2,1,1,0,272,1952,656,75,10,2,1,1,0,1385,17408,7442,816,75,10,2,
%U A321400 1,1,0,7936,177280,99280,11407,832,75,10,2,1,1
%N 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.
%C A321400 See the comments and definitions in A320956. Note also the corresponding construction for the exp function in A320955.
%e A321400 Array starts:
%e A321400 n\k   0  1  2   3   4    5      6       7        8  ...
%e A321400 -------------------------------------------------------
%e A321400   [0] 1, 0, 0,  0,  0,   0,     0,      0,       0, ... A000007
%e A321400   [1] 1, 1, 1,  2,  5,  16,    61,    272,    1385, ... A000111
%e A321400   [2] 1, 1, 2,  8, 40, 256,  1952,  17408,  177280, ... A000828
%e A321400   [3] 1, 1, 2, 10, 70, 656,  7442,  99280, 1515190, ... A320957
%e A321400   [4] 1, 1, 2, 10, 75, 816, 11407, 194480, 3871075, ... A321394
%e A321400   [5] 1, 1, 2, 10, 75, 832, 12322, 232560, 5325325, ...
%e A321400   [6] 1, 1, 2, 10, 75, 832, 12383, 238272, 5693735, ...
%e A321400   [7] 1, 1, 2, 10, 75, 832, 12383, 238544, 5732515, ...
%e A321400   [8] 1, 1, 2, 10, 75, 832, 12383, 238544, 5733900, ...
%e A321400 -------------------------------------------------------
%e A321400 Seen as a triangle given by descending antidiagonals:
%e A321400   [0] 1
%e A321400   [1] 0,  1
%e A321400   [2] 0,  1,   1
%e A321400   [3] 0,  1,   1,  1
%e A321400   [4] 0,  2,   2,  1,  1
%e A321400   [5] 0,  5,   8,  2,  1, 1
%e A321400   [6] 0, 16,  40, 10,  2, 1, 1
%e A321400   [7] 0, 61, 256, 70, 10, 2, 1, 1
%p A321400 sf := proc(n) option remember; `if`(n <= 1, 1-n, (n-1)*(sf(n-1) + sf(n-2))) end:
%p A321400 kernel := proc(n, k) option remember; binomial(n, k)*sf(k) end:
%p A321400 egf := n -> add(kernel(n, k)*((tan + sec)(x*(n - k))), k=0..n):
%p A321400 A321400Row := proc(n, len) series(egf(n), x, len + 2):
%p A321400 seq(coeff(%, x, k)*k!/n!, k=0..len) end:
%p A321400 seq(lprint(A321400Row(n, 9)), n=0..9);
%Y A321400 Cf. A000111 (n=1), A000828 (n=2), A320957 (n=3), A321394 (n=4), A320956 (limit).
%Y A321400 Antidiagonal sums (and row sums of the triangle): A321399.
%K A321400 nonn,tabl
%O A321400 0,12
%A A321400 _Peter Luschny_, Nov 08 2018