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A274742 Triangle read by rows: T(n,k) (n>=3, 0<=k<=n-3) = number of n-sequences of 0's and 1's that begin with 1 and have exactly one pair of adjacent 0's and exactly k pairs of adjacent 1's. 1
1, 1, 1, 2, 2, 1, 2, 4, 3, 1, 3, 6, 6, 4, 1, 3, 9, 12, 8, 5, 1, 4, 12, 18, 20, 10, 6, 1, 4, 16, 30, 30, 30, 12, 7, 1, 5, 20, 40, 60, 45, 42, 14, 8, 1, 5, 25, 60, 80, 105, 63, 56, 16, 9, 1, 6, 30, 75, 140, 140, 168, 84, 72, 18, 10, 1, 6, 36, 105, 175, 280, 224, 252, 108, 90, 20, 11, 1, 7, 42, 126, 280, 350, 504, 336, 360, 135, 110, 22, 12, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
3,4
COMMENTS
It appears that the row sums give the positive integers of A001629. - Omar E. Pol, Jul 09 2016
LINKS
FORMULA
T(n,k) = binomial(floor((n+k-2)/2),k)*floor((n-k-1)/2).
EXAMPLE
n=3 => 100 -> T(3,0) = 1.
n=4 => 1001 -> T(4,0) = 1; 1100 -> T(4,1) = 1.
n=5 => 10010, 10100 -> T(5,0) = 1; 10011, 11001 -> T(5,1) = 2;
11100 -> T(5,2) = 1.
Triangle starts:
1
1, 1
2, 2, 1
2, 4, 3, 1
3, 6, 6, 4, 1
3, 9, 12, 8, 5, 1
4, 12, 18, 20, 10, 6, 1
4, 16, 30, 30, 30, 12, 7, 1
5, 20, 40, 60, 45, 42, 14, 8, 1
5, 25, 60, 80, 105, 63, 56, 16, 9, 1
6, 30, 75, 140, 140, 168, 84, 72, 18, 10, 1
6, 36, 105, 175, 280, 224, 252, 108, 90, 20, 11, 1
7, 42, 126, 280, 350, 504, 336, 360, 135, 110, 22, 12, 1
MATHEMATICA
Table[Binomial[Floor[(n + k - 2)/2], k] Floor[(n - k - 1)/2], {n, 3, 15}, {k, 0, n - 3}] // Flatten (* Michael De Vlieger, Jul 05 2016 *)
PROG
(PARI) t(n, k) = binomial(floor((n+k-2)/2), k) * floor((n-k-1)/2)
trianglerows(n) = for(x=3, n+2, for(y=0, x-3, print1(t(x, y), ", ")); print(""))
trianglerows(13) \\ Felix Fröhlich, Jul 05 2016
CROSSREFS
Columns: A008619, A087811.
Sequence in context: A353368 A300667 A129687 * A128176 A368415 A370075
KEYWORD
nonn,tabl
AUTHOR
Jeremy Dover, Jul 04 2016
STATUS
approved

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Last modified March 29 06:57 EDT 2024. Contains 371265 sequences. (Running on oeis4.)