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A218225 G.f. A(x) satisfies: (1 - x*A(x)) / (1 - x^2*A(x)^2)^2  =  1 - x. 1
1, 2, 6, 23, 101, 480, 2400, 12434, 66142, 359112, 1981904, 11085198, 62696874, 357970472, 2060459256, 11943445311, 69656978837, 408466559630, 2406825745010, 14243262687023, 84618295006269, 504485687485408, 3017344000161296, 18099717207764928 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Binomial transform of A001002. - Vladimir Kruchinin, Oct 03 2014

Conjecture: a(n) is the number of permutations of [1..n+1] that avoid one of the following sets of patterns: (2134, 42153, 24153), (3124, 42153, 24153), (2143, 42135, 24135). - Alexander Burstein, Dec 20 2017

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.

FORMULA

G.f. A(x) satisfies:

(1) A(1 - (1-x)/(1-x^2)^2)  =  x + 1/(1-x-x^2).

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

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

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

Recurrence: 5*n*(n+1)*a(n) = 21*n*(2*n-1)*a(n-1) - 3*(23*n^2-46*n+24)*a(n-2) + 16*(n-2)*(2*n-3)*a(n-3). - Vaclav Kotesovec, May 22 2013

a(n) ~ 2^(5*n+6)/(27*sqrt(Pi)*5^(n+1/2)*n^(3/2)). - Vaclav Kotesovec, May 22 2013

G.f.: (-1+cos(2/3*(arccot(3*sqrt(3/5))-arccot((3*sqrt(3))/sqrt(5-32*x))))+sqrt(15)*sin(2/3*(arccot(3*sqrt(3/5))-arccot((3*sqrt(3))/sqrt(5-32*x)))))/(3*x). - Vaclav Kotesovec, Jul 06 2013

EXAMPLE

G.f.: A(x) = 1 + 2*x + 6*x^2 + 23*x^3 + 101*x^4 + 480*x^5 + 2400*x^6 +...

The series reversion of x*A(x) begins:

x - 2*x^2 + 2*x^3 - 3*x^4 + 3*x^5 - 4*x^6 + 4*x^7 - 5*x^8 + 5*x^9 +...

so that  A(1 - (1-x)/(1-x^2)^2) = x + 1/(1-x-x^2).

The logarithm of the g.f. equals the series:

log(A(x)) = (1 + (1-x)*A(x))*x +

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

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

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

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

MATHEMATICA

Table[1/(n+1)*SeriesCoefficient[(((x-1)*(x+1)^2)/(x^2+x-1))^(n+1), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 22 2013 *)

Flatten[{1, Table[FullSimplify[SeriesCoefficient[(2*(1-x)^(1/3)-2^(2/3)*(-11-16*x-3*Sqrt[-15+96*x])^(1/3)+2^(2/3)*(11+16*x-3*Sqrt[-15+96*x])^(1/3))/(6*(-1+x)^(1/3)*x), {x, 0, n}]], {n, 1, 10}]}] (* Vaclav Kotesovec, Jul 06 2013 *)

CoefficientList[Series[(-1+Cos[2/3*(ArcCot[3*Sqrt[3/5]]-ArcCot[(3*Sqrt[3])/Sqrt[5-32*x]])]+Sqrt[15]*Sin[2/3*(ArcCot[3*Sqrt[3/5]]-ArcCot[(3*Sqrt[3])/Sqrt[5-32*x]])])/(3*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 06 2013 *)

PROG

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

for(n=0, 30, print1(a(n), ", "))

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

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Sequence in context: A078487 A193038 A213090 * A279572 A263576 A231444

Adjacent sequences:  A218222 A218223 A218224 * A218226 A218227 A218228

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Oct 23 2012

STATUS

approved

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Last modified September 18 20:10 EDT 2021. Contains 347534 sequences. (Running on oeis4.)