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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.

%I A274742 #22 Jul 10 2016 18:26:22
%S A274742 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,
%T A274742 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,
%U A274742 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
%N 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.
%C A274742 It appears that the row sums give the positive integers of A001629. - _Omar E. Pol_, Jul 09 2016
%F A274742 T(n,k) = binomial(floor((n+k-2)/2),k)*floor((n-k-1)/2).
%e A274742 n=3 => 100 -> T(3,0) = 1.
%e A274742 n=4 => 1001 -> T(4,0) = 1; 1100 -> T(4,1) = 1.
%e A274742 n=5 => 10010, 10100 -> T(5,0) = 1; 10011, 11001 -> T(5,1) = 2;
%e A274742        11100 -> T(5,2) = 1.
%e A274742 Triangle starts:
%e A274742 1
%e A274742 1, 1
%e A274742 2, 2, 1
%e A274742 2, 4, 3, 1
%e A274742 3, 6, 6, 4, 1
%e A274742 3, 9, 12, 8, 5, 1
%e A274742 4, 12, 18, 20, 10, 6, 1
%e A274742 4, 16, 30, 30, 30, 12, 7, 1
%e A274742 5, 20, 40, 60, 45, 42, 14, 8, 1
%e A274742 5, 25, 60, 80, 105, 63, 56, 16, 9, 1
%e A274742 6, 30, 75, 140, 140, 168, 84, 72, 18, 10, 1
%e A274742 6, 36, 105, 175, 280, 224, 252, 108, 90, 20, 11, 1
%e A274742 7, 42, 126, 280, 350, 504, 336, 360, 135, 110, 22, 12, 1
%t A274742 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 *)
%o A274742 (PARI) t(n, k) = binomial(floor((n+k-2)/2), k) * floor((n-k-1)/2)
%o A274742 trianglerows(n) = for(x=3, n+2, for(y=0, x-3, print1(t(x, y), ", ")); print(""))
%o A274742 trianglerows(13) \\ _Felix Fröhlich_, Jul 05 2016
%Y A274742 Cf. A046854, A274228.
%Y A274742 Columns: A008619, A087811.
%K A274742 nonn,tabl
%O A274742 3,4
%A A274742 _Jeremy Dover_, Jul 04 2016