0,4

We may also relabel the entries as U(0,0), U(1,0), U(0,1), U(2,0), U(1,1), U(0,2), U(3,0), ...

U(n,k) is the number of ways of writing the vector (n,k) as an ordered sum of vectors, equivalently, the number of paths from (0,0) to (n,k) in which steps may be taken from (i,j) to (p,q) provided (p,q) is to the right or above (i,j). - Jon Stadler (jstadler(AT)capital.edu), Apr 30 2003

2*U(n,k) = SUM_{i<=n,j<=k} U(i,j) - Jon Stadler (jstadler(AT)capital.edu), Apr 30 2003

U(n,k) = 2U(n-1,k) + SUM_{i<k} U(n,i) - Jon Stadler (jstadler(AT)capital.edu), Apr 30 2003

U(n,k) = SUM_{0<=j<=n+k} C(n,j-k+1)*C(k,j-n+1)*2^j - Jon Stadler (jstadler(AT)capital.edu), Apr 30 2003

U(n,k) = 0 if k<0 or k>n; else 1 if n <= 1; else 3 if n=2 and k=1; else 2U(n-1,k-1) + 2U(n-1,k) - 2U(n-2,k-1). - David W. Wilson (davidwwilson(AT)comcast.net)

Index entries for triangles and arrays related to Pascal's triangle

G.f. U(z, w) = Sum_{n >= 0, k >= 0} U(n, k)*z^n*w^k = Sum{n >= 0, k >= 0} T(n, k)*z^(n-k)*w^k = (1-z)*(1-w)/(1-2*w-2*z+2*z*w).

Maple code gives another explicit formula for U(n, k).

T(n, k)=2*(T(n-1, k-1)+T(n-1, k))-(2-0^(n-2))*T(n-2, k-1) for n>1 and 1<k<n, T(n, 0)=T(n, n)=2*T(n-1, 0) for n>0, T(0, 0)=1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 03 2004

1; 1,1; 2,3,2; 4,8,8,4; ...

T(5,2) is the sum of the elements above it in the parallelogram bordered by T(0,0), T(3,0), T(2,2) and T(5,2).

A059576 := proc(n, k) local b, t1; t1 := min(n+k-2, n, k); add( (-1)^b * 2^(n+k-b-2) * (n+k-b-2)! * (1/(b! * (n-b)! * (k-b)!)) * (-2 * n-2 * k+2 * k^2+b^2-3 * k * b+2 * n^2+5 * n * k-3 * n * b), b=0..t1); end;

Cf. A035002, A059226, A008288, A059283.

First diagonals give A000079, A001792. T(2n, n) gives A052141. Row sums give A003480.

Sequence in context: A108838 A105070 A154578 this_sequence A034800 A082771 A127157

Adjacent sequences: A059573 A059574 A059575 this_sequence A059577 A059578 A059579

easy,nonn,tabl,nice

Floor van Lamoen (fvlamoen(AT)hotmail.com), Jan 23 2001