A168592 - OEIS

%I #31 Jun 17 2024 07:15:55

%S 1,3,14,80,509,3459,24579,180389,1356743,10402493,81004516,638886082,

%T 5093081983,40971735401,332187974718,2711668091448,22267979870143,

%U 183830653156341,1524747465249750,12700172705956876,106187411693668179

%N G.f.: exp( Sum_{n>=1} A082758(n)*x^n/n ), where A082758(n) = sum of the squares of the trinomial coefficients in row n of triangle A027907.

%C Number of lattice paths from (0,0) to (n,n) which do not go above the diagonal x=y using steps (1,k), (k,1) with k >= 0 and two kinds of (1,1). - _Alois P. Heinz_, Oct 07 2015

%C Number of pairs of noncrossing paths of length n which start and end together, each taking steps (1,0), (1,1) or (1,-1) (i.e., Motzkin-type). - _Nicholas R. Beaton_, Jun 17 2024

%H Alois P. Heinz, <a href="/A168592/b168592.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: A(x) = (1/x)*Series_Reversion[x*(1-x)^2/((1+x)^2*(1-x+x^2))].

%F G.f.: A(x) satisfies A(x^2) = M(x)*M(-x), where M(x) is the g.f. of A001006. - _Alexander Burstein_, Oct 03 2017

%F G.f.: A(x) satisfies A(x^2) = (1-x - sqrt(1-2*x-3*x^2))*(1+x - sqrt(1+2*x-3*x^2))/(4*x^4). - _Paul D. Hanna_, Oct 05 2017, concluded from formula of Alexander Burstein.

%e G.f.: A(x) = 1 + 3*x + 14*x^2 + 80*x^3 + 509*x^4 + 3459*x^5 + ...

%e log(A(x)) = 3*x + 19*x^2/2 + 141*x^3/3 + 1107*x^4/4 + 8953*x^5/5 + ... + A082758(n)*x^n/n + ...

%p b:= proc(x, y) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,

%p       add(b(x-i, y-1), i=0..x) +add(b(x-1, y-j), j=0..y)))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Oct 07 2015

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n<4, [1, 3, 14, 80][n+1],

%p       ((10*(n+1))*(16*n^3-20*n^2-n-1) *a(n-1)

%p       +(-944*n^4+2596*n^3-1924*n^2+236*n+30) *a(n-2)

%p       +(90*(n-2))*(16*n^3-52*n^2+45*n-6) *a(n-3)

%p       -(81*(2*n-5))*(n-2)*(n-3)*(4*n-1) *a(n-4))/

%p       ((n+1)*(4*n-5)*(2*n+1)*(n+2)))

%p     end:

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Oct 07 2015

%t (1/x)*InverseSeries[x*(1 - x)^2/((1 + x)^2*(1 - x + x^2)) + O[x]^30, x] // CoefficientList[#, x]& (* _Jean-François Alcover_, Jun 09 2018 *)

%o (PARI) {a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sum(k=0,2*m, polcoeff((1+x+x^2)^m,k)^2)*x^m/m) +x*O(x^n)),n))}

%o (PARI) {a(n)=polcoeff(1/x*serreverse(x*(1-x)^2/((1+x)^2*(1-x+x^2)+x*O(x^n))),n)}

%Y Cf. A168590, A168593, A082758, A027907, A168595, A218321, A263316.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 01 2009