A276313
Number of weak up-down sequences of length n and values in {1,2,...,n}.
Original entry on oeis.org
1, 1, 3, 14, 85, 671, 6405, 72302, 940005, 13846117, 227837533, 4142793511, 82488063476, 1785049505682, 41715243815059, 1046997553798894, 28089178205661221, 802173732190546289, 24296253228394108980, 777918130180655893150, 26253270588637259772768
Offset: 0
a(0) = 1: the empty sequence.
a(1) = 1: 1.
a(2) = 3: 11, 12, 22.
a(3) = 14: 111, 121, 122, 131, 132, 133, 221, 222, 231, 232, 233, 331, 332, 333.
a(4) = 85: 1111, 1112, 1113, 1114, 1211, ..., 4423, 4424, 4433, 4434, 4444.
-
b:= proc(n, k, t) option remember; `if`(n=0, 1,
add(b(n-1, k, k-j), j=1..t))
end:
a:= n-> b(n, n+1, n):
seq(a(n), n=0..25);
-
b[n_, k_, t_] := b[n, k, t] = If[n==0, 1, Sum[b[n-1, k, k-j], {j, 1, t}]];
a[n_] := b[n, n+1, n];
Table[a[n], {n, 0, 25}](* Jean-François Alcover, May 18 2017, translated from Maple *)
A373423
Array read by ascending antidiagonals: T(n, k) = [x^k] cf(n) where cf(0) = 1, cf(1) = -1/(x - 1), and for n > 1 is cf(n) = ~( ~x - 1/(~x - 1/(~x - 1/(~x - 1/(~x - ... 1/(~x + 1))))...) ) where '~' is '-' if n is even, and '+' if n is odd, and x appears n times in the expression.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 1, 1, 0, 1, 4, 3, 1, 1, 0, 1, 5, 6, 5, 1, 1, 0, 1, 6, 10, 14, 8, 1, 1, 0, 1, 7, 15, 30, 31, 13, 1, 1, 0, 1, 8, 21, 55, 85, 70, 21, 1, 1, 0, 1, 9, 28, 91, 190, 246, 157, 34, 1, 1, 0, 1, 10, 36, 140, 371, 671, 707, 353, 55, 1, 1, 0
Offset: 0
Generating functions of row n:
gf0 = 1;
gf1 = - 1/( x-1);
gf2 = x + 1/(-x+1);
gf3 = x - 1/( x-1/( x+1));
gf4 = x + 1/(-x-1/(-x-1/(-x+1)));
gf5 = x - 1/( x-1/( x-1/( x-1/( x+1))));
gf6 = x + 1/(-x-1/(-x-1/(-x-1/(-x-1/(-x+1)))));
.
Array begins:
[0] 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
[1] 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
[2] 1, 2, 1, 1, 1, 1, 1, 1, 1, ... A373565
[3] 1, 3, 3, 5, 8, 13, 21, 34, 55, ... A373566
[4] 1, 4, 6, 14, 31, 70, 157, 353, 793, ... A373567
[5] 1, 5, 10, 30, 85, 246, 707, 2037, 5864, ... A373568
[6] 1, 6, 15, 55, 190, 671, 2353, 8272, 29056, ... A373569
A000217, A006322, A108675, ...
A000330, A085461, A244881, ...
.
Triangle starts:
[0] 1;
[1] 1, 0;
[2] 1, 1, 0;
[3] 1, 2, 1, 0;
[4] 1, 3, 1, 1, 0;
[5] 1, 4, 3, 1, 1, 0;
[6] 1, 5, 6, 5, 1, 1, 0;
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row := proc(n, len) local x, a, j, ser;
if n = 0 then a := -1 elif n = 1 then a := -1/(x - 1) elif irem(n, 2) = 1 then
a := x + 1; for j from 1 to n-1 do a := x - 1 / a od: else
a := -x + 1; for j from 1 to n-1 do a := -x - 1 / a od: fi;
ser := series((-1)^(n-1)*a, x, len + 2); seq(coeff(ser, x, j), j = 0..len) end:
A := (n, k) -> row(n, 12)[k+1]: # array form
T := (n, k) -> row(n - k, k+1)[k+1]: # triangular form
seq(lprint([n], row(n, 9)), n = 0..9);
-
def Arow(n, len):
R. = PowerSeriesRing(ZZ, len)
if n == 0: return [1] + [0]*(len - 1)
if n == 1: return [1]*(len - 1)
x = x if n % 2 == 1 else -x
a = x + 1
for _ in range(n - 1):
a = x - 1 / a
if n % 2 == 0: a = -a
return a.list()
for n in range(8): print(Arow(n, 9))
Showing 1-2 of 2 results.