A092566 Main diagonal of triangle A092565, in which the n-th row polynomial equals the numerator of the n-th convergent of the continued fraction [1 + x + x^2; 1 + x + x^2, 1 + x + x^2, ...].

1, 1, 3, 7, 22, 63, 191, 573, 1752, 5372, 16597, 51465, 160258, 500551, 1567881, 4922687, 15488481, 48821964, 154147654, 487412324, 1543231353, 4891986889, 15524303265, 49314008259, 156791992914, 498931763064, 1588891019625

Offset: 0

Author

Paul D. Hanna, Feb 28 2004

Keywords

• nonn
• nice

Comments

T(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (1,0), (2,0), (1,1), and (1,2). - Joerg Arndt, Jun 30 2011

Diagonal of rational function 1/(1 - (x + x^2 + x*y + x*y^2)). - Gheorghe Coserea, Aug 06 2018

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1959

Crossrefs

Cf. A092565, A037027.

Programs

Maple

series(RootOf((27*x^4-14*x^3+9*x^2+14*x-5)*y^3+(4-3*x)*y+1, y), x=0, 30); # Mark van Hoeij, Apr 16 2013

Mathematica

A037027[n_, k_] := Sum[Binomial[k+j, k]*Binomial[j, n-j-k], {j, 0, n-k}]; A037027[n_, 0] = Fibonacci[n+1]; a[n_] := Sum[A037027[n, k]*Binomial[k, n-k], {k, 0, n}]; Table[a(n), {n,0,26}] (* Jean-François Alcover, Jul 18 2011 *)
a[0, 0] = 1; a[n_, k_] /; n >= 0 && k >= 0 := a[n, k] = a[n, k-1] + a[n, k-2] + a[n-1, k-1] + a[n-2, k-1]; a[, ] = 0;
a[n_] := a[n, n];
a /@ Range[0, 30] (* Jean-François Alcover, Oct 06 2019, after Joerg Arndt *)

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(nA037027(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 */
/* Joerg Arndt, Jun 30 2011 */

Formula

a(n) = sum(k=0..n, A037027(n, k)*C(k, n-k) ).

O.g.f. A(x) satisfies the equation (27*x^4 - 14*x^3 + 9*x^2 + 14*x - 5)*A(x)^3 + (4-3*x)*A(x) + 1 = 0. - Mark van Hoeij, Apr 16 2013