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A216617 G.f. satisfies: A(x) = (1 + x*(1-x)*A(x)^2) * (1 + x^2*A(x)). 4
1, 1, 2, 5, 13, 37, 112, 351, 1130, 3716, 12424, 42101, 144277, 499136, 1740871, 6114629, 21609654, 76786625, 274171192, 983187372, 3539498904, 12787269117, 46345303727, 168463177245, 614002351108, 2243406499930, 8215549186628, 30149687633264, 110861650218443 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Apparently the number of Dyck paths of semilength n that avoid UUDUUD. The only Dyck path of semilength 4 that contains UUDUUD is UUDUUDdd. So a(4) = A000108(4)-1 = 13. - David Scambler, Apr 24 2013

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n,k)^2 * x^k*(1-x)^(n-k) * A(x)^(n-k) ).

Recurrence: (n+1)*(n+2)*(679*n^4 - 7380*n^3 + 23045*n^2 - 9120*n - 29484)*a(n) = (n+1)*(1358*n^5 - 14081*n^4 + 33259*n^3 + 60800*n^2 - 249924*n + 117936)*a(n-1) + (7469*n^6 - 81180*n^5 + 247067*n^4 - 13482*n^3 - 719746*n^2 + 667728*n - 176904)*a(n-2) - 2*(5432*n^6 - 67188*n^5 + 260696*n^4 - 210849*n^3 - 724651*n^2 + 1374444*n - 589500)*a(n-3) + 6*(1358*n^6 - 18834*n^5 + 89081*n^4 - 133447*n^3 - 140626*n^2 + 464272*n - 173424)*a(n-4) - 2*(9506*n^6 - 146097*n^5 + 784589*n^4 - 1442697*n^3 - 1099897*n^2 + 5950320*n - 4023900)*a(n-5) + 2*(6790*n^6 - 114540*n^5 + 682867*n^4 - 1465407*n^3 - 769658*n^2 + 6637308*n - 5733000)*a(n-6) + 2*(n-6)*(n-5)*(1358*n^4 - 10007*n^3 + 8773*n^2 + 34598*n - 27540)*a(n-7) - 4*(n-7)*(n-6)*(679*n^4 - 4664*n^3 + 4979*n^2 + 17546*n - 22260)*a(n-8). - Vaclav Kotesovec, Sep 16 2013

a(n) ~ c*d^n/n^(3/2), where d = 3.8781907052914131... is the root of the equation 4 + 4*d - 16*d^2 - 8*d^3 - 12*d^4 + d^6 = 0 and c = 0.561628033... - Vaclav Kotesovec, Sep 16 2013

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 37*x^5 + 112*x^6 + 351*x^7 +...

The logarithm of the g.f. begins:

log(A(x)) = ((1-x)*A(x) + x)*x + ((1-x)^2*A(x)^2 + 2^2*x*(1-x)*A(x) + x^2)*x^2/2 +

((1-x)^3*A(x)^3 + 3^2*x*(1-x)^2*A(x)^2 + 3^2*x^2*(1-x)*A(x) + x^3)*x^3/3 +

((1-x)^4*A(x)^4 + 4^2*x*(1-x)^3*A(x)^3 + 6^2*x^2*(1-x)^2*A(x)^2 + 4^2*x^3*(1-x)*A(x) + x^4)*x^4/4 +

((1-x)^5*A(x)^5 + 5^2*x*(1-x)^4*A(x)^4 + 10^2*x^2*(1-x)^3*A(x)^3 + 10^2*x^3*(1-x)^2*A(x)^2 + 5^2*x^4*(1-x)*A(x) + x^5)*x^5/5 +...

Explicitly,

log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 31*x^4/4 + 106*x^5/5 + 378*x^6/6 + 1359*x^7/7 + 4935*x^8/8 + 18073*x^9/9 + 66578*x^10/10 +...

MAPLE

a:= n->coeff(series(RootOf(A=(1+x*(1-x)*A^2)*(1+x^2*A), A), x, n+1), x, n):

seq(a(n), n=0..30);  # Alois P. Heinz, Apr 25 2013

MATHEMATICA

m = 30; A[_] = 0;

Do[A[x_] = (1 + x (1 - x) A[x]^2) (1 + x^2 A[x]) + O[x]^m, {m}];

CoefficientList[A[x], x] (* Jean-Fran├žois Alcover, Oct 02 2019 *)

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1 + x*(1-x)*A^2)*(1+x^2*A) +x*O(x^n)); polcoeff(A, n)}

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n+1, x^m/m*sum(k=0, m, binomial(m, k)^2*x^k*(1-x)^(m-k)*A^(m-k) +x*O(x^n))))); polcoeff(A, n)}

for(n=0, 40, print1(a(n), ", "))

CROSSREFS

Cf. A216616, A216604, A216447.

Sequence in context: A151442 A263529 A053732 * A243412 A170941 A119495

Adjacent sequences:  A216614 A216615 A216616 * A216618 A216619 A216620

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 10 2012

STATUS

approved

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Last modified September 25 06:32 EDT 2020. Contains 337335 sequences. (Running on oeis4.)