A188442 Expansion of -(sqrt(-3*x^2-2*x+1)-x-1)/(2*sqrt(-3*x^2-2*x+1)+2*x).

0, 1, 1, 3, 6, 16, 39, 103, 269, 721, 1941, 5285, 14476, 39918, 110633, 308081, 861390, 2417264, 6805477, 19216297, 54404562, 154402264, 439172189, 1251701837, 3574234089, 10223988665, 29292665059, 84052426683, 241518890124, 694900077226

Offset: 0

Author

Vladimir Kruchinin, Apr 20 2011

Keywords

• nonn

Comments

Apparently for n>0 the number of grand Motzkin paths of length n-1 that avoid flat steps at level 1. - David Scambler, Jul 04 2013

a(n) is the number of increasing unary-binary trees with associated permutation that simultaneously avoids 213 and 312. For more information about increasing unary-binary trees with an associated permutation, see A245888. - Manda Riehl, Aug 22 2014

Crossrefs

A000984 is the sequence of the odd-indexed terms of a(n).

Programs

Mathematica

a[n_] := Sum[ k*Fibonacci[k]*Sum[ Binomial[-k+2*j-1, j-1]*(-1)^(n-j)*Binomial[n, j], {j, k, n}], {k, 1, n}]/n; a[0] = 0; Table[a[n], {n, 0, 29}] (* Jean-François Alcover, Jun 14 2013, translated from Maxima *)

Maxima

a(n):=sum(k*fib(k)*sum(binomial(-k+2*j-1,j-1)*(-1)^(n-j)*binomial(n,j),j,k,n),k,1,n)/n;

PARI

x='x+O('x^66);
gf=-(sqrt(-3*x^2-2*x+1)-x-1)/(2*sqrt(-3*x^2-2*x+1)+2*x); /* = 0 +x +x^2 +3*x^3 +... */
Vec(gf) /* Joerg Arndt, Apr 21 2011 */

Formula

G.f.: A000045(x*A005043(x))

a(n)=sum(k=1..n,k*A000045(k)*sum(j=k..n, binomial(-k+2*j-1,j-1)*(-1)^(n-j)*binomial(n,j)))/n.

a(n)=sum(k=1..n, T097609(n,k)*A000045(k)).

D-finite with recurrence n*(5*n-2)*a(n) +(-20*n^2+23*n-12)*a(n-1) +3*(-5*n^2+7*n+20)*a(n-2) +2*(35*n^2-89*n-12)*a(n-3) +12*(5*n+3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 23 2017