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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 Cf. A046854, A274228. Columns: A008619, A087811. Sequence in context: A275297 A300667 A129687 * A128176 A144963 A305632 Adjacent sequences:  A274739 A274740 A274741 * A274743 A274744 A274745 KEYWORD nonn,tabl AUTHOR Jeremy Dover, Jul 04 2016 STATUS approved

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Last modified December 8 20:37 EST 2021. Contains 349596 sequences. (Running on oeis4.)