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A073157 Number of Schroeder n-paths containing no FFs. 10
1, 2, 5, 18, 70, 293, 1283, 5808, 26960, 127628, 613814, 2990681, 14730713, 73229291, 366936231, 1851352820, 9397497758, 47957377934, 245903408244, 1266266092112, 6545667052320, 33954266444498, 176689391245146 (list; graph; refs; listen; history; text; internal format)
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: A322555 A118814 A014271 * A268570 A141494 A189843

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

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Last modified March 22 10:07 EDT 2019. Contains 321421 sequences. (Running on oeis4.)