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A120588 G.f. is 1 + x*c(x), where c(x) is the g.f. of the Catalan numbers (A000108). 26
1, 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, 18367353072152 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Previous name was: G.f. satisfies: 3*A(x) = 2 + x + A(x)^2, with A(0) = 1.

This is essentially a duplicate of entry A000108, the Catalan numbers (a(n) = A000108(n-1) for n>0).

In order for the g.f. of an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n, where n > 1, it is necessary that the sequence start with [1, d, m*n*(n-1)/2], where d divides m*n*(n-1)/2 (m>0) and that the coefficients are given by r = n + d^2/m, c = r-1 and b = d^3/m. The remaining terms may then be integer and still satisfy: a_n(k) = r*a(k), where a_n(k) is the k-th term of the n-th self-convolution of the sequence.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: A(x) = 1 + Series_Reversion(1+3*x - (1+x)^2).

Lagrange Inversion yields g.f.: A(x) = Sum_{n>=0} C(2*n,n)/(n+1)*(2+x)^(n+1)/3^(2*n+1).

G.f.: (3 - sqrt(1-4*x))/2. - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009

a(n) = Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011

G.f.: 2 - G(0), where G(k)= 2*x*(2*k+1) + k +1 - 2*x*(k+1)*(2*k+3)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 14 2013

G.f.: 2 - G(0), where G(k)= 1 - x/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Jul 19 2013

a(n) ~ 2^(2*n-2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 19 2013

Given g.f. A(x), A001850(n-1) = coefficient of x^n in A(x)^n if n>0, the derivative of log(A(x)) is the g.f. for A026641. - Michael Somos, May 18 2015

A(x) = (1 + 2*Sum_{n >= 1} Catalan(n)*x^n)/(1 + Sum_{n >= 1} Catalan(n)*x^n) = (1 + 3/2*Sum_{n >= 1} binomial(2*n,n)*x^n )/(1 + Sum_{n >= 1} binomial(2*n,n)*x^n). - Peter Bala, Sep 01 2016

EXAMPLE

A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 +...

A(x)^3 = 1 + 2*x + 3*x^2 + 6*x^3 + 15*x^4 + 42*x^5 + 126*x^6 + 396*x^7 +..

More generally, given the functional equation:

r*A(x) = r-1 + b*x + A(x)^n

the series solution is:

A(x) = Sum_{i>=0} C(n*i,i)/(n*i-i+1)*(r-1+bx)^(n*i-i+1)/r^(n*i+1)

which can be expressed as:

A(x) = G( (r-1+bx)^(n-1)/r^n ) * (r-1+bx)/r

where G(x) satisfies: G(x) = 1 + x*G(x)^n .

Also we have:

A(x) = 1 + Series_Reversion[ (1 + r*x - (1+x)^n )/b ].

MATHEMATICA

a[ n_] := SeriesCoefficient[ 1 + (1 - Sqrt[1 - 4 x]) / 2, {x, 0, n}]; (* Michael Somos, May 18 2015 *)

PROG

(PARI) {a(n)=local(A=1+x+x^2+x*O(x^n)); for(i=0, n, A=A-3*A+2+x+A^2); polcoeff(A, n)}

(PARI) {a(n) = my(A); if( n<1, n==0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 23 2011 */

(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (3 - Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 18 2019

(Sage) ((3-sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 18 2019

CROSSREFS

Cf. A120589 (A(x)^2); A120590 - A120607.

Cf. A001850, A026641.

Sequence in context: A291825 A287974 A115140 * A168491 A000108 A057413

Adjacent sequences:  A120585 A120586 A120587 * A120589 A120590 A120591

KEYWORD

nonn,easy

AUTHOR

Paul D. Hanna, Jun 16 2006, Jan 24 2008

EXTENSIONS

New name by Wolfdieter Lang, Feb 06 2020

STATUS

approved

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Last modified October 6 08:17 EDT 2022. Contains 357263 sequences. (Running on oeis4.)