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A112806 Expansion of solution of functional equation. 2
1, 1, 2, 6, 21, 79, 312, 1277, 5369, 23049, 100612, 445214, 1992606, 9004260, 41025315, 188259072, 869305315, 4036286518, 18832973733, 88259024068, 415252542641, 1960718710035, 9288106921038, 44129146527731 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..23.

Gi-Sang Cheon, S.-T. Jin, L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Available online 30 March 2015.

FORMULA

Given g.f. A(x), then series reversion of B(x)=x*A(x^3) is -B(-x).

Given g.f. A(x), then y=x*A(x^3) satisfies y=x+(xy)^2/(1-(xy)^3).

G.f. satisfies: A(x) = 1 + x*A(x)^2/(1 - x^2*A(x)^3). - Paul D. Hanna, Jun 06 2012

G.f. satisfies: A(x) = 1/A(-x*A(x)^3); note that the Catalan function C(x) = 1 + x*C(x)^2 (A000108) also satisfies this condition. - Paul D. Hanna, Jun 06 2012

a(n) = Sum_{i=0..n/2}((binomial(n+2*i+1,i)*Sum_{k=0..n-2*i}(binomial(k,n-k-2*i)*(-1)^(n-k)*binomial(n+k+2*i,k)))/(n+2*i+1)). - Vladimir Kruchinin, Mar 07 2016

PROG

(PARI) {a(n)=local(A); if(n<0, 0, A=x+O(x^4); for(k=1, n, A=x+subst(x^2/(1-x^3), x, x*A)); polcoeff(A, 3*n+1))}

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+x*A^2/(1-x^2*A^3)); polcoeff(A, n)} \\ Paul D. Hanna, Jun 06 2012

(Maxima)

a(n):=sum((binomial(n+2*i+1, i)*sum(binomial(k, n-k-2*i)*(-1)^(n-k)*binomial(n+k+2*i, k), k, 0, n-2*i))/(n+2*i+1), i, 0, n/2); /* Vladimir Kruchinin, Mar 07 2016 */

CROSSREFS

Cf. A004148, A216490.

Sequence in context: A257562 A033321 A050203 * A150199 A150200 A150201

Adjacent sequences:  A112803 A112804 A112805 * A112807 A112808 A112809

KEYWORD

nonn

AUTHOR

Michael Somos, Sep 20 2005

STATUS

approved

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Last modified May 9 13:09 EDT 2021. Contains 343742 sequences. (Running on oeis4.)