(PARI) a(n)=if(n<0, 0, polcoeff(contfracpnqn(vector(n, i, 1+x+x^2))[1, 1], n, x))
(PARI)
A037027(n, k)=if(n<k || k<0, 0, sum(j=0, n-k, binomial(j+k, k)*binomial(j, n- j-k)))
a(n)=sum(k=0, n,
A037027(n, k)*binomial(k, n-k))
(PARI) /* computation as lattice paths: */
N=40; /* that many terms */
B=matrix(N, N); B[1, 1]=1; /* whether T(n, k) memoized */
M=matrix(N, N); M[1, 1]=1; /* memoization for T(n, k) */
steps=[[1, 0], [2, 0], [1, 1], [1, 2]];
T(n, k)=
{
my(ret, dx, dy);
if ( n<0, return(0) );
if ( k<0, return(0) );
if ( B[n+1, k+1], return( M[n+1, k+1]) );
ret = 0;
for (s=1, #steps,
dx = steps[s][1];
dy = steps[s][2];
ret += T( n-dx, k-dy );
);
B[n+1, k+1] = 1;
M[n+1, k+1] = ret;
return( ret );
}
T(N-1, N-1); /* trigger computations */
for (n=1, N, print1(M[n, n], ", ")); /* show (diagonal) terms */
for(n=0, N-1, for(k=0, n, print1(T(n, k), ", "); ); print(); ); /* show triangle */
