A073157 Number of Schroeder n-paths containing no FFs.

1, 2, 5, 18, 70, 293, 1283, 5808, 26960, 127628, 613814, 2990681, 14730713, 73229291, 366936231, 1851352820, 9397497758, 47957377934, 245903408244, 1266266092112, 6545667052320, 33954266444498, 176689391245146

Offset

0,2

Comments

Number of Schroeder n-paths containing no FFs. A Schroeder n-path (A006318) consists of steps U=(1,1),F=(2,0),D=(1,-1) starting at (0,0), ending at (2n,0), and never going below the x-axis. Example: a(2)=5 counts UFD, UUDD, UDF, FUD, UDUD. - David Callan, Aug 23 2011

Links

Muniru A Asiru, Table of n, a(n) for n = 0..300

Formula

A073155(n+1) = Sum_{k=0..n} a(k)*a(n-k), that is, convolution yields sequence A073155 minus the 0th term.

G.f.: A(x) = (1 - sqrt(1 - 4*x*(1+x)^2))/(2*x*(1+x)) satisfies A(x) = (1+x)*(1 + x*A(x)^2);

G.f.: A(x) = (1+x)*C(x*(1+x)^2) where C(x) is the Catalan g.f. of A000108. - Paul D. Hanna, Mar 03 2008

a(n) = Sum_{j=0..n}((Sum_{k=0..j}((binomial(2*k+2,j-k)*C(k))))*(-1)^(n-j)), where C(k) = A000108(k). - Vladimir Kruchinin, Mar 13 2016

a(n) = Sum_{i=0..n} C(2*i+1,i)*C(2*i+1,n-i))/(2*i+1). - Vladimir Kruchinin, Oct 11 2018

Recurrence: (n+1)*a(n) = 3*(n-1)*a(n-1) + 6*(2*n - 3)*a(n-2) + 6*(2*n - 5)*a(n-3) + 2*(2*n - 7)*a(n-4). - Vaclav Kotesovec, Oct 11 2018

Example

G.f.: A(x) = 1 + 2*x + 5*x^2 + 18*x^3 + 70*x^4 + 293*x^5 + 1283*x^6 + ...

Maple

a:=n->add(binomial(2*i+1, i)*binomial(2*i+1, n-i)/(2*i+1), i=0..n): seq(a(n), n=0..25); # Muniru A Asiru, Oct 11 2018

Mathematica

Table[Sum[Binomial[2*i + 1, i]*Binomial[2*i + 1, n - i]/(2*i + 1), {i, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Oct 11 2018 *)

Prog

(PARI) {a(n)=local(A=1); for(i=0, n-1, A=(1+x)*(1+x*(A+x*O(x^n))^2)); polcoeff(A, n)} /* Paul D. Hanna, Mar 03 2008 */
(Maxima)
a(n):=sum((sum((binomial(2*k+2, j-k)*binomial(2*k, k)/(k+1)), k, 0, j))*(-1)^(n-j), j, 0, n); /* Vladimir Kruchinin, Mar 13 2016 */
(GAP) List([0..25], n->Sum([0..n], i->Binomial(2*i+1, i)*Binomial(2*i+1, n-i)/(2*i+1))); # Muniru A Asiru, Oct 11 2018

Crossrefs

Leftmost column of triangle A073154 (was previous name).

Cf. A073155, A073156, A073153, A000108.

Sequence in context: A118814 A345878 A014271 * A365120 A268570 A141494

Adjacent sequences: A073154 A073155 A073156 * A073158 A073159 A073160

Keyword

easy,nonn

Author

Paul D. Hanna, Jul 29 2002

Extensions

More terms from Paul D. Hanna, Mar 03 2008

New name using a comment from David Callan, Peter Luschny, Oct 14 2018

Status

approved