login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A182309 Triangle T(n,k) with 2 <= k <= floor(2(n+1)/3) gives the number of length-n binary sequences with exactly k zeros and with length two for the longest run of zeros. 0
1, 2, 3, 2, 4, 6, 1, 5, 12, 6, 6, 20, 18, 3, 7, 30, 40, 16, 1, 8, 42, 75, 50, 10, 9, 56, 126, 120, 45, 4, 10, 72, 196, 245, 140, 30, 1, 11, 90, 288, 448, 350, 126, 15, 12, 110, 405, 756, 756, 392, 90, 5, 13, 132, 550, 1200, 1470, 1008, 357, 50, 1, 14, 156, 726 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
COMMENTS
Triangle T(n,k) captures several well known sequences. In particular, T(n,2)=(n-1), the natural numbers; T(n,3)=(n-2)(n-3)=A002378(n-3), the "oblong" numbers; T(n,4)=(n-3)(n-4)^2/2=A002411(n-4), "pentagonal pyramidal" numbers; and also T(n,5)=(n-4)C(n-4,3)=A004320(n-6). Furthermore, row sums=A000100(n+1).
LINKS
FORMULA
T(n,k) = Sum_{j=1..k/2} binomial(n-k+1,j)*binomial(n-k-j+1,k-2j) for 2 <= k <= 2(n+1)/3.
EXAMPLE
For n=6 and k=3, T(6,3)=12 since there are 12 binary sequences of length 6 that contain 3 zeros and that have a maximum run of zeros of length 2, namely, 011100, 101100, 110100, 011001, 101001, 110010, 010011, 100110, 100101, 001110, 001101, and 001011.
Triangle T(n,k) begins
1,
2,
3, 2,
4, 6, 1,
5, 12, 6,
6, 20, 18, 3,
7, 30, 40, 16, 1,
8, 42, 75, 50, 10,
9, 56, 126, 120, 45, 4,
10, 72, 196, 245, 140, 30, 1,
11, 90, 288, 448, 350, 126, 15,
12, 110, 405, 756, 756, 392, 90, 5,
13, 132, 550, 1200, 1470, 1008, 357, 50, 1,
14, 156, 726, 1815, 2640, 2268, 1106, 266, 21,
15, 182, 936, 2640, 4455, 4620, 2898, 1016, 161, 6,
MAPLE
seq(seq(sum(binomial(n-k+1, j)*binomial(n-k+1-j, k-2*j), j=1..floor(k/2)), k=2..floor(2*(n+1)/3)), n=2..20);
MATHEMATICA
t[n_, k_] := Sum[ Binomial[n-k+1, j]*Binomial[n-k-j+1, k-2*j], {j, 1, k/2}]; Table[t[n, k], {n, 2, 15}, {k, 2, 2*(n+1)/3}] // Flatten (* Jean-François Alcover, Jun 06 2013 *)
CROSSREFS
Row sums of triangle T(n,k)=A000100(n+1);
T(n,3)=A002378(n-3); T(n,4)=A002411(n-4);
T(n,5)=A004320(n-6).
Sequence in context: A325349 A320054 A215228 * A043263 A321326 A118978
KEYWORD
nonn,nice,easy,tabf
AUTHOR
Dennis P. Walsh, Apr 23 2012
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)