A276312
Number of up-down sequences of length n and values in {1,2,...,n}.
Original entry on oeis.org
1, 1, 1, 5, 31, 246, 2353, 26585, 345775, 5094220, 83833256, 1524414737, 30353430420, 656851828075, 15350023574061, 385261255931365, 10335781852020335, 295166535640444376, 8939894824857438940, 286234265613041061128, 9659753724363828753408
Offset: 0
a(0) = 1: the empty sequence.
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 5: 121, 131, 132, 231, 232.
a(4) = 31: 1212, 1213, 1214, 1312, 1313, 1314, 1323, 1324, 1412, 1413, 1414, 1423, 1424, 1434, 2312, 2313, 2314, 2323, 2324, 2412, 2413, 2414, 2423, 2424, 2434, 3412, 3413, 3414, 3423, 3424, 3434.
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b:= proc(n, k, t) option remember; `if`(n=0, 1,
add(b(n-1, k, k-j), j=1..t-1))
end:
a:= n-> b(n, n+1$2):
seq(a(n), n=0..25);
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b[n_, k_, t_] := b[n, k, t] = If[n==0, 1, Sum[b[n-1, k, k-j], {j, 1, t-1}]];
a[n_] := b[n, n+1, n+1];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)
A373424
Array read by ascending antidiagonals: T(n, k) = [x^k] cf(n) where cf(n) is the continued fraction (-1)^n/(~x - 1/(~x - ... 1/(~x - 1)))...) and 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, 3, 1, 0, 1, 4, 6, 5, 1, 0, 1, 5, 10, 14, 8, 1, 0, 1, 6, 15, 30, 31, 13, 1, 0, 1, 7, 21, 55, 85, 70, 21, 1, 0, 1, 8, 28, 91, 190, 246, 157, 34, 1, 0, 1, 9, 36, 140, 371, 671, 707, 353, 55, 1, 0, 1, 10, 45, 204, 658, 1547, 2353, 2037, 793, 89, 1, 0
Offset: 0
Generating functions of the rows:
gf0 = 1;
gf1 = -1/( x-1);
gf2 = 1/(-x-1/(-x-1));
gf3 = -1/( x-1/( x-1/( x-1)));
gf4 = 1/(-x-1/(-x-1/(-x-1/(-x-1))));
gf5 = -1/( x-1/( x-1/( x-1/( x-1/( x-1)))));
gf6 = 1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1))))));
...
Array A(n, k) starts:
[0] 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... A000007
[1] 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A000012
[2] 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... A000045
[3] 1, 3, 6, 14, 31, 70, 157, 353, 793, 1782, ... A006356
[4] 1, 4, 10, 30, 85, 246, 707, 2037, 5864, 16886, ... A006357
[5] 1, 5, 15, 55, 190, 671, 2353, 8272, 29056, 102091, ... A006358
[6] 1, 6, 21, 91, 371, 1547, 6405, 26585, 110254, 457379, ... A006359
A000027,A000330, A085461, A244881, ...
A000217, A006322, A108675, ...
.
Triangle T(n, k) = A(n - k, k) starts:
[0] 1;
[1] 1, 0;
[2] 1, 1, 0;
[3] 1, 2, 1, 0;
[4] 1, 3, 3, 1, 0;
[5] 1, 4, 6, 5, 1, 0;
[6] 1, 5, 10, 14, 8, 1, 0;
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row := proc(n, len) local x, a, j, ser; if irem(n, 2) = 1 then
a := x - 1; for j from 1 to n do a := x - 1 / a od: a := a - x; else
a := -x - 1; for j from 1 to n do a := -x - 1 / a od: a := -a - x;
fi; ser := series(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
-
def Arow(n, len):
R. = PowerSeriesRing(ZZ, len)
if n == 0: return [1] + [0]*(len - 1)
x = -x if n % 2 else x
a = x + 1
for _ in range(n):
a = x - 1 / a
a = x - a if n % 2 else a - x
return a.list()
for n in range(7): print(Arow(n, 10))
Showing 1-2 of 2 results.
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