A047072 Array A read by diagonals: A(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 step touches the line y=x unless x=0 or x=h.
1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 2, 2, 3, 1, 1, 4, 5, 4, 5, 4, 1, 1, 5, 9, 5, 5, 9, 5, 1, 1, 6, 14, 14, 10, 14, 14, 6, 1, 1, 7, 20, 28, 14, 14, 28, 20, 7, 1, 1, 8, 27, 48, 42, 28, 42, 48, 27, 8, 1, 1, 9, 35, 75, 90, 42, 42, 90, 75, 35, 9, 1
Offset: 0
Examples
Array, A(n, k), begins as: 1, 1, 1, 1, 1, 1, 1, 1, ...; 1, 2, 1, 2, 3, 4, 5, 6, ...; 1, 1, 2, 2, 5, 9, 14, 20, ...; 1, 2, 2, 4, 5, 14, 28, 48, ...; 1, 3, 5, 5, 10, 14, 42, 90, ...; 1, 4, 9, 14, 14, 28, 42, 132, ...; 1, 5, 14, 28, 42, 42, 84, 132, ...; 1, 6, 20, 48, 90, 132, 132, 264, ...; Antidiagonals, T(n, k), begins as: 1; 1, 1; 1, 2, 1; 1, 1, 1, 1; 1, 2, 2, 2, 1; 1, 3, 2, 2, 3, 1; 1, 4, 5, 4, 5, 4, 1; 1, 5, 9, 5, 5, 9, 5, 1; 1, 6, 14, 14, 10, 14, 14, 6, 1;
Links
- G. C. Greubel, Antidiagonals n = 0..50, flattened
- R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
Crossrefs
Programs
-
Magma
b:= func< n | n eq 0 select 1 else 2*Catalan(n-1) >; function A(n,k) if k eq n then return b(n); elif k gt n then return Binomial(n+k-1, n) - Binomial(n+k-1, n-1); else return Binomial(n+k-1, k) - Binomial(n+k-1, k-1); end if; return A; end function; // [[A(n,k): k in [0..12]]: n in [0..12]]; T:= func< n,k | A(n-k, k) >; [T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 13 2022
-
Mathematica
A[, 0]= 1; A[0, ]= 1; A[h_, k_]:= A[h, k]= If[(k-1>h || k-1
-
SageMath
def A(n,k): if (k==n): return 2*catalan_number(n-1) + 2*int(n==0) elif (k>n): return binomial(n+k-1, n) - binomial(n+k-1, n-1) else: return binomial(n+k-1, k) - binomial(n+k-1, k-1) def T(n,k): return A(n-k, k) # [[A(n,k) for k in range(12)] for n in range(12)] flatten([[T(n,k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Oct 13 2022
Formula
A(n, n) = 2*[n=0] - A002420(n),
A(n, n+1) = 2*A000108(n-1), n >= 1.
From G. C. Greubel, Oct 13 2022: (Start)
T(n, n-1) = A000027(n-2) + 2*[n<3], n >= 1.
T(n, n-2) = A000096(n-4) + 2*[n<5], n >= 2.
T(n, n-3) = A005586(n-6) + 4*[n<7] - 2*[n=3], n >= 3.
T(2*n, n) = 2*A000108(n-1) + 3*[n=0].
T(2*n-1, n-1) = T(2*n+1, n+1) = A000180(n).
T(3*n, n) = A025174(n) + [n=0]
Sum_{k=0..n} T(n, k) = 2*A063886(n-2) + [n=0] - 2*[n=1]
Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n, k) = A047079(n). (End)