login
Array read by diagonals: T(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and no up-step crosses the line y=4x/3. (Thus a path crosses the line only at lattice points and on right-steps.).
11

%I #11 Jan 19 2020 20:38:41

%S 1,1,1,1,2,1,1,1,3,1,1,2,4,4,1,1,3,2,8,5,1,1,4,5,10,13,6,1,1,5,9,15,

%T 23,19,7,1,1,6,14,24,38,42,26,8,1,1,7,20,38,62,80,68,34,9,1,1,8,27,58,

%U 38,142,148,102,43,10,1,1,9,35,85,96,180,290

%N Array read by diagonals: T(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and no up-step crosses the line y=4x/3. (Thus a path crosses the line only at lattice points and on right-steps.).

%H Andrew Howroyd, <a href="/A047140/b047140.txt">Table of n, a(n) for n = 0..1325</a>

%e Array begins:

%e ======================================

%e h\k | 0 1 2 3 4 5 6 7

%e ----+---------------------------------

%e 0 | 1 1 1 1 1 1 1 1 ...

%e 1 | 1 2 1 2 3 4 5 6 ...

%e 2 | 1 3 4 2 5 9 14 20 ...

%e 3 | 1 4 8 10 15 24 38 58 ...

%e 4 | 1 5 13 23 38 62 38 96 ...

%e 5 | 1 6 19 42 80 142 180 96 ...

%e 6 | 1 7 26 68 148 290 470 566 ...

%e 7 | 1 8 34 102 250 540 1010 1576 ...

%e ...

%o (PARI) A(h,k=h)={my(M=matrix(h+1,k+1,i,j,1)); for(h=1, h, for(k=1, k, M[1+h, 1+k] = M[h,1+k] + if(3*k>4*h && 3*(k-1)<4*h, 0, M[1+h,k]))); M}

%o { my(T=A(10)); for(i=1, #T, print(T[i,]))} \\ _Andrew Howroyd_, Jan 19 2020

%Y Cf. A047141, A047142, A047143, A047144, A047145, A047146, A047147, A047148, A047149.

%Y Cf. A047110, A047130, A047150.

%K nonn,tabl

%O 0,5

%A _Clark Kimberling_. Definition revised Dec 08 2006