|
| |
|
|
A078531
|
|
Coefficients of power series that satisfies A(x)^2 - 4x*A(x)^3 = 1, A(0)=1.
|
|
5
| |
|
|
1, 2, 10, 64, 462, 3584, 29172, 245760, 2124694, 18743296, 168043980, 1526726656, 14025209100, 130056978432, 1215785268840, 11445014102016, 108401560073190, 1032295389593600, 9877854438949980, 94927710773575680
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| If A(x)=sum_{k=1..inf} a(k)x^k satisfies A(x)^n - (n^2)*x*A(x)^(n+1) = 1, then a(k)=n^(2k)*binomial(k/n+1/n+k-1,k)/(k+1) and, consequently, a(n-1) = n^(2n-3) and a(2n-1) = n^(4n-2). - Emeric Deutsch, Dec 10 2002
A generalization of the Catalan sequence (A000108) since for n = 1 the equation A(x)^n -(n^2)*x*A(x)^(n+1) = 1 reduces to A(x)=1+xA(x)^2. - Emeric Deutsch, Dec 10 2002
Number of symmetric non-crossing connected graphs on 2n+1 equidistant nodes on a circle (it is assumed that the axis of symmetry is a diameter of the circle passing through a given node). Example: a(1)=2 because on the nodes A,B,C (axis of symmetry through A) the only symmetric non-crossing connected graphs are {AB,AC} and {AB,AC,BC}. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 03 2003
|
|
|
REFERENCES
| P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 1999, 203-229.
|
|
|
FORMULA
| a(n)=2*sum(binomial(3n-3, i)*binomial(2n-2-i, n), i=0..n-2)/(n-1) for n>1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 29 2002
G.f.: (12x)^(-1) + (6x)^(-1)*sin(arcsin(216x^2-1)/3). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 30 2002
a(n)=2^(2n)*binomial(3n/2-1/2, n)/(n+1) - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 10 2002
G.f. A(x)=y satisfies y'(6xy-1)+2y^2=0, y'(y^2-3)+4y^4=0. - Michael Somos Feb 05 2004
Sequence with offset 1 is expansion of reversion of g.f. x*sqrt(1-4x). - R. Stephan, Mar 22 2004
G.f. satisfies: A(x) = Sum_{n>=0} ((2n)!/n!^2)*x^n*A(x)^n. [Paul D. Hanna, Mar 3 2011]
|
|
|
EXAMPLE
| A(x)^2 - 4x*A(x)^3 = 1 since A(x)^2 = 1 + 4x + 24x^2 + 148x^3 + 1280x^4 + 10296x^5 + ... and A(x)^3 = 1 + 6x + 42x^2 + 320x^3 + 2574x^4 + ... also a(1)=2^1, a(3)=2^6.
|
|
|
PROG
| (PARI) a(n)=if(n<0, 0, n++; polcoeff(serreverse(x*sqrt(1-4*x+O(x^n))), n)) - Michael Somos Feb 05 2004
(PARI) a(n)=if(n<1, n==0, polcoeff(serreverse(x*(2+x)/(4*(1+x)^3)+x*O(x^n)), n)) - Michael Somos Feb 05 2004
(PARI) {a(n)=local(B=sum(m=0, n, binomial(2*m, m)*x^m+x*O(x^n))); polcoeff(1/x*serreverse(x/B), n)} [Paul D. Hanna, Mar 3 2011]
|
|
|
CROSSREFS
| Cf. A078532, A078533, A078534, A078535.
Sequence in context: A183165 A129130 A186268 * A130721 A167449 A064170
Adjacent sequences: A078528 A078529 A078530 * A078532 A078533 A078534
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Nov 28 2002
|
| |
|
|
