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Array read by descending antidiagonals: T(h,k) is the 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 = 3x/4. (Thus a path crosses the line only at lattice points and on right-steps.)
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%I #14 Jan 20 2020 09:20:47

%S 1,1,1,1,1,1,1,2,2,1,1,3,2,3,1,1,4,5,5,4,1,1,5,9,5,9,5,1,1,6,14,14,14,

%T 14,6,1,1,7,20,28,28,28,20,7,1,1,8,27,48,56,28,48,27,8,1,1,9,35,75,

%U 104,84,76,75,35,9,1,1,10,44,110,179,188,84

%N Array read by descending antidiagonals: T(h,k) is the 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 = 3x/4. (Thus a path crosses the line only at lattice points and on right-steps.)

%H Andrew Howroyd, <a href="/A047130/b047130.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 1 2 3 4 5 6 7 ...

%e 2 | 1 2 2 5 9 14 20 27 ...

%e 3 | 1 3 5 5 14 28 48 75 ...

%e 4 | 1 4 9 14 28 56 104 179 ...

%e 5 | 1 5 14 28 28 84 188 367 ...

%e 6 | 1 6 20 48 76 84 272 639 ...

%e 7 | 1 7 27 75 151 235 272 911 ...

%e ...

%o (PARI)

%o 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(4*k>3*h && 4*(k-1)<3*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. A047131, A047132, A047133, A047134, A047135, A047136, A047137, A047138, A047139.

%Y Cf. A047110, A047140, A047150.

%K nonn,tabl

%O 0,8

%A _Clark Kimberling_. Definition revised Dec 08 2006